6879:
713:
consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the
7898:
8996:
In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These
7475:
9864:
7376:
7289:
4984:
6681:
6635:
10595:
8754:
12129:
8991:
4789:
772:
to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity. In 1876,
8377:
13801:
9469:
9143:
8906:
7663:
7784:
9275:
7390:
8010:
11093:
8854:
8488:
10180:
3926:
10111:
8116:
4024:
10010:
7201:
4332:
9790:
6310:
4253:
10981:
10936:
10248:
13830:
11134:
8546:
7941:
7300:
7213:
12373:
12052:
11691:
11645:
7712:
7067:
4122:
12328:
11971:
11597:
7608:
6983:
6874:{\displaystyle {\mathfrak {a}}{\mathfrak {b}}:=\{a_{1}b_{1}+\dots +a_{n}b_{n}\mid a_{i}\in {\mathfrak {a}}{\mbox{ and }}b_{i}\in {\mathfrak {b}},i=1,2,\dots ,n;{\mbox{ for }}n=1,2,\dots \},}
6673:
4810:
3964:
1895:
7100:
4410:
244:
12890:
12842:
6552:
4174:
13809:
9705:
9084:
536:
13772:
5174:
11485:
11400:
10669:
9958:
9907:
9307:
4522:
3281:
5509:
5254:
1837:
11863:
11776:
11317:
11043:
10886:
10707:
10478:
10401:
8552:
4445:
10468:
10347:
3796:
3123:
682:
or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these
13306:
12724:
12684:
12290:
12228:
12192:
12161:
11547:
9393:
11833:
11746:
489:
452:
12425:
12399:
12254:
12006:
11806:
11719:
11008:
10833:
10802:
10774:
10731:
10622:
10425:
10310:
9355:
9331:
7965:
7767:
7737:
7550:
7526:
7502:
7150:
7124:
7031:
7007:
6944:
6914:
6538:
6512:
4080:
4056:
3420:
3008:
2847:
2255:
1807:
13201:
9662:
8259:
3356:
13276:
10058:
9614:
3749:
2761:
2570:
2360:
2311:
13332:
13249:
13225:
12786:
12472:
12059:
11187:
10856:
8046:
2914:
2892:
2487:
2410:
2384:
2284:
198:
8157:
12752:
8911:
4373:
3839:
1577:
3501:
2789:
2703:
2598:
1732:
5367:
3652:
3213:
2677:
1378:
1241:
11262:
11225:
5333:
4658:
4556:
4479:
3454:
3172:
2454:
2013:
1971:
1206:
1156:
1125:
6160:
6132:
5058:
4281:
3089:
3054:
2975:
2091:
1923:
1539:
1501:
1475:
1024:
996:
6106:
5650:
5433:
5389:
2286:
of all integers, since the sum of any two even integers is even, and the product of any integer with an even integer is also even; this ideal is usually denoted by
1697:
965:
927:
787:, to which Dedekind had added many supplements. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by
9744:
9576:
9534:
9501:
9203:
3604:
3567:
3538:
2173:
2120:
9776:
5200:
5114:
5086:
1770:
1668:
1603:
1341:
1284:
5795:
5021:
4635:
4606:
2056:
1445:
1049:
13838:
13166:
13144:
13124:
13102:
13080:
13060:
13040:
13018:
12992:
12972:
12952:
12916:
12644:
12620:
12593:
12571:
12549:
12526:
12494:
9166:
8199:
8177:
6348:
6230:
5881:
5859:
5839:
5817:
5770:
5748:
5728:
5704:
5676:
5553:
5531:
5455:
5411:
5305:
5283:
4205:
3714:
3694:
3674:
3378:
3303:
3028:
2938:
2870:
2809:
2725:
2641:
2621:
2534:
2512:
2333:
2193:
2142:
2033:
1071:
892:
870:
848:
1409:
By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a
8266:
9403:
9096:
8860:
7617:
7893:{\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}}
7558:: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a
1624:(For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.)
8014:. It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the
7470:{\displaystyle {\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})\supset {\mathfrak {a}}\cap {\mathfrak {b}}+{\mathfrak {a}}\cap {\mathfrak {c}}}
9208:
7972:
5567:
To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.
13375:
13424:
does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in
11057:
8760:
8387:
641:
13780:
10116:
9536:
is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals.
3847:
10067:
3800:. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the
9859:{\displaystyle \operatorname {nil} (R)=\bigcap _{{\mathfrak {p}}{\text{ prime ideals }}}{\mathfrak {p}}\subset \operatorname {Jac} (R)}
8054:
3977:
9965:
7162:
4288:
6249:
4560:, in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see
4210:
13660:
10942:
10897:
10187:
13722:
11100:
7371:{\displaystyle ({\mathfrak {a}}+{\mathfrak {b}}){\mathfrak {c}}={\mathfrak {a}}{\mathfrak {c}}+{\mathfrak {b}}{\mathfrak {c}}}
7284:{\displaystyle {\mathfrak {a}}({\mathfrak {b}}+{\mathfrak {c}})={\mathfrak {a}}{\mathfrak {b}}+{\mathfrak {a}}{\mathfrak {c}}}
13979:
13946:
8493:
7912:
9020:
For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings.
98:
12335:
12013:
11652:
11606:
13880:
7671:
7038:
12995:
12918:
is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term
6393:
Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:
4979:{\displaystyle RXR=\{r_{1}x_{1}s_{1}+\dots +r_{n}x_{n}s_{n}\mid n\in \mathbb {N} ,r_{i}\in R,s_{i}\in R,x_{i}\in X\}.\,}
7569:. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a
4087:
12297:
1507:(say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring
14027:
14001:
13908:
13889:
13732:
11942:
11566:
8015:
7579:
6954:
6644:
3935:
1842:
777:
replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of
727:
634:
586:
7074:
783:
14100:
6630:{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}:=\{a+b\mid a\in {\mathfrak {a}}{\mbox{ and }}b\in {\mathfrak {b}}\}}
6058:: An ideal is said to be irreducible if it cannot be written as an intersection of ideals that properly contain it.
4567:
An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subset
4378:
208:
12851:
12803:
4136:
9669:
9051:
5575:. Different types of ideals are studied because they can be used to construct different types of factor rings.
503:
6386:
5121:
14090:
14085:
13926:
11409:
11324:
10631:
9931:
9880:
9280:
4484:
10590:{\displaystyle {\mathfrak {a}}^{e}={\Big \{}\sum y_{i}f(x_{i}):x_{i}\in {\mathfrak {a}},y_{i}\in B{\Big \}}}
8749:{\displaystyle {\mathfrak {a}}{\mathfrak {b}}=(z(x+z),z(y+w),w(x+z),w(y+w))=(z^{2}+xz,zy+wz,wx+wz,wy+w^{2})}
14095:
3656:, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of
3222:
627:
494:
17:
5464:
5207:
4805:
is defined in the similar way. For "two-sided", one has to use linear combinations from both sides; i.e.,
1812:
13355:
13350:
11837:
11750:
11271:
11017:
10860:
10681:
10375:
6322:
4419:
344:
10440:
10319:
3756:
3096:
686:
and absorption properties are the defining properties of an ideal. An ideal can be used to construct a
14019:
13283:
12691:
12651:
12265:
12203:
12168:
12136:
11494:
9366:
104:
11818:
11731:
465:
428:
119:
12406:
12380:
12235:
12124:{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}=B_{\mathfrak {p}}\Rightarrow {\mathfrak {p}}^{e}}
11987:
11787:
11700:
10989:
10814:
10783:
10755:
10712:
10603:
10406:
10291:
9710:
9336:
9312:
7946:
7748:
7718:
7531:
7507:
7483:
7131:
7105:
7012:
6988:
6925:
6895:
6519:
6493:
6039:
6027:
4061:
4037:
3842:
3385:
2982:
2821:
2198:
1788:
723:
13178:
9621:
8209:
3312:
3010:
is given by those functions that vanish for large enough arguments, i.e. those continuous functions
1774:. Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.
13256:
10017:
9593:
8986:{\displaystyle {\mathfrak {a}}\cap {\mathfrak {c}}=(w,xz+z^{2})\neq {\mathfrak {a}}{\mathfrak {c}}}
7775:(More generally, the difference between a product and an intersection of ideals is measured by the
3719:
2917:
2734:
2543:
2340:
2291:
579:
382:
332:
13315:
13232:
13208:
12759:
12445:
11139:
10841:
8029:
5571:
Ideals are important because they appear as kernels of ring homomorphisms and allow one to define
2897:
2875:
2461:
2393:
2367:
2267:
181:
11978:
8124:
391:
125:
84:
12731:
4784:{\displaystyle RX=\{r_{1}x_{1}+\dots +r_{n}x_{n}\mid n\in \mathbb {N} ,r_{i}\in R,x_{i}\in X\}.}
4352:
3818:
1544:
3467:
2768:
2682:
2577:
2413:
1704:
548:
399:
350:
131:
5340:
3625:
3186:
2646:
1349:
1214:
11234:
11192:
5312:
4529:
4458:
3427:
3139:
2426:
1980:
1938:
1185:
1132:
1101:
683:
6139:
6111:
5028:
4260:
3059:
3033:
2945:
2061:
1902:
1518:
1480:
1454:
1003:
975:
14037:
13956:
13365:
12926:
9870:
7566:
6241:
6197:
6079:
6047:
5629:
5418:
5374:
2600:
1675:
1609:
1247:
938:
900:
742:
706:
272:
146:
14045:
12898:
is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When
9720:
9552:
9510:
9477:
9179:
3580:
3543:
3514:
2149:
2096:
1608:
The notion of an ideal does not involve associativity; thus, an ideal is also defined for
8:
11559:
9755:
9540:
6237:
5897:
5609:
5262:
5179:
5093:
5065:
4638:
2850:
1749:
1647:
1582:
1318:
1301:
1261:
554:
362:
313:
258:
152:
138:
34:
5777:
5003:
4617:
4588:
2038:
1427:
1031:
13989:
13718:
13345:
13151:
13129:
13109:
13087:
13065:
13045:
13025:
13003:
12977:
12957:
12937:
12901:
12629:
12605:
12578:
12556:
12534:
12511:
12479:
11913:
9151:
8372:{\displaystyle {\mathfrak {a}}=(z,w),{\mathfrak {b}}=(x+z,y+w),{\mathfrak {c}}=(x+z,w)}
8184:
8162:
6382:
6333:
6215:
6008:. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
5866:
5844:
5824:
5802:
5755:
5733:
5713:
5689:
5661:
5538:
5516:
5440:
5396:
5290:
5268:
4561:
4190:
3699:
3679:
3659:
3363:
3288:
3013:
2923:
2855:
2794:
2710:
2626:
2606:
2519:
2497:
2387:
2318:
2178:
2127:
2018:
1305:
1056:
877:
855:
833:
804:
667:
567:
53:
9015:
Ideals appear naturally in the study of modules, especially in the form of a radical.
14023:
13997:
13975:
13942:
13904:
13885:
13728:
12501:
10285:
9910:
9585:
9010:
6054:
3216:
1410:
1391:
1255:
608:
405:
170:
111:
13992:; Gubareni, Nadiya; Gubareni, Nadezhda MikhaΔlovna; Kirichenko, Vladimir V. (2004).
9464:{\displaystyle J=\{x\in R\mid 1-yx\,{\text{ is a unit element for every }}y\in R\}.}
9138:{\displaystyle J=\bigcap _{{\mathfrak {m}}{\text{ maximal ideals}}}{\mathfrak {m}}.}
5265:(equivalence relations that respect the ring structure) on the ring: Given an ideal
14041:
13934:
13875:
11897:
11563:
10269:
9046:
7740:
7559:
6433:
6398:
6359:
1395:
1175:. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".
1168:
774:
750:
614:
600:
414:
356:
319:
92:
78:
8901:{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}
7658:{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}
4610:. Such an ideal exists since it is the intersection of all left ideals containing
14033:
13971:
13952:
13930:
11881:
9030:
7906:
7562:
6192:
6184:
6034:
6012:
5902:
5890:
4346:
4178:
1398:. Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore,
1288:
731:
710:
695:
376:
326:
164:
11049:. Many classic examples of this stem from algebraic number theory. For example,
14067:
13918:
13871:
13676:
7384:
If a product is replaced by an intersection, a partial distributive law holds:
6407:
6002:
5946:
4031:
827:
699:
420:
13938:
14079:
12438:
11925:
11809:
10358:
9925:
9038:
6441:
is defined as a fractional ideal for which there is another fractional ideal
6315:
6204:
6168:
6043:
5957:
5616:
5580:
3176:. Thus, a skew-field is simple and a simple commutative ring is a field. The
1782:
1670:
since it is precisely the two-sided ideal generated (see below) by the unity
788:
735:
687:
561:
457:
72:
13852:
13360:
9504:
9270:{\displaystyle R/\operatorname {Ann} (M)=R/\operatorname {Ann} (x)\simeq M}
5950:
1777:
An (left, right or two-sided) ideal that is not the unit ideal is called a
1503:. (Right and two-sided ideals are defined similarly.) For a ring, an ideal
792:
769:
765:
753:
is a generalization of an ideal, and the usual ideals are sometimes called
746:
719:
691:
679:
593:
368:
264:
14062:
8005:{\displaystyle {\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}}
6421:
with a special property. If the fractional ideal is contained entirely in
5620:: A nonzero ideal is called minimal if it contains no other nonzero ideal.
14011:
13380:
11937:
11722:
7776:
6459:. Some authors may also apply "invertible ideal" to ordinary ring ideals
5682:
5605:
5572:
4996:
is called the principal left (resp. right, two-sided) ideal generated by
3177:
3131:
1613:
1088:
715:
659:
655:
573:
284:
158:
40:
13527:
Fermat's last theorem. A genetic introduction to algebraic number theory
11088:{\displaystyle \mathbb {Z} \to \mathbb {Z} \left\lbrack i\right\rbrack }
8849:{\displaystyle {\mathfrak {a}}{\mathfrak {c}}=(xz+z^{2},zw,xw+zw,w^{2})}
13963:
13370:
8998:
8483:{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(z,w,x+z,y+w)=(x,y,z,w)}
5886:
2420:
2315:. More generally, the set of all integers divisible by a fixed integer
1932:
1388:
338:
10175:{\displaystyle J\cdot ({\mathfrak {a}}/\operatorname {Ann} (J^{n}))=0}
3921:{\displaystyle \operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}}
1809:
is proper if and only if it does not contain a unit element, since if
12623:
12529:
11050:
10106:{\displaystyle {\mathfrak {a}}\supsetneq \operatorname {Ann} (J^{n})}
6176:
4416:, then there is an ideal that is maximal among the ideals containing
4127:
1080:
778:
298:
203:
13988:
6884:
i.e. the product is the ideal generated by all products of the form
4992:
A left (resp. right, two-sided) ideal generated by a single element
27:
Additive subgroup of a mathematical ring that absorbs multiplication
7570:
2386:
is generated by its smallest positive element, as a consequence of
1400:
the two-sided ideals are exactly the kernels of ring homomorphisms.
930:
823:
292:
278:
9474:
For a not-necessarily-commutative ring, it is a general fact that
9397:. There is also another characterization (the proof is not hard):
8111:{\displaystyle (n)\cap (m)=\operatorname {lcm} (n,m)\mathbb {Z} }
4019:{\displaystyle ({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}}}
2261:
1171:, the three definitions are the same, and one talks simply of an
675:
176:
60:
6180:: An ideal is a nil ideal if each of its elements is nilpotent.
12497:
10005:{\displaystyle \operatorname {nil} (R)=\operatorname {Jac} (R)}
7196:{\displaystyle {\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}}
4327:{\displaystyle \textstyle \bigcup _{i\in S}{\mathfrak {a}}_{i}}
705:
Among the integers, the ideals correspond one-for-one with the
671:
14018:. Annals of Mathematics Studies. Vol. 72. Princeton, NJ:
6305:{\displaystyle {\textrm {grade}}(I)={\textrm {proj}}\dim(R/I)}
4248:{\displaystyle {\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j}}
894:
is a left ideal if it satisfies the following two conditions:
13617:
13615:
13613:
13611:
10976:{\displaystyle {\mathfrak {b}}^{ce}\subseteq {\mathfrak {b}}}
10931:{\displaystyle {\mathfrak {a}}^{ec}\supseteq {\mathfrak {a}}}
10243:{\displaystyle J^{n}{\mathfrak {a}}=J^{n+1}{\mathfrak {a}}=0}
2456:
is an ideal in the ring of all real-coefficient polynomials
2423:
with real coefficients that are divisible by the polynomial
1973:
are its only ideals and conversely: that is, a nonzero ring
13903:(Third ed.). Hoboken, NJ: John Wiley & Sons, Inc.
9090:
is the intersection of all primitive ideals. Equivalently,
6490:
The sum and product of ideals are defined as follows. For
1927:. Typically there are plenty of proper ideals. In fact, if
13699:
13687:
13639:
13608:
12551:
that "absorbs multiplication from the left by elements of
11129:{\displaystyle B=\mathbb {Z} \left\lbrack i\right\rbrack }
9543:) is built-in to the definition of a Jacobson radical: if
6172:: This term has multiple uses. See the article for a list.
3611:
850:
that "absorbs multiplication from the left by elements of
674:
of its elements. Ideals generalize certain subsets of the
13923:
Commutative
Algebra with a View toward Algebraic Geometry
6985:
is the smallest left (resp. right) ideal containing both
4797:(since such a span is the smallest left ideal containing
10255:
3134:
if it is nonzero and has no two-sided ideals other than
14063:"The Geometric Interpretation for Extension of Ideals?"
13486:
has a unit, this extra requirement becomes superfluous.
8541:{\displaystyle {\mathfrak {a}}+{\mathfrak {c}}=(z,w,x)}
5261:
There is a bijective correspondence between ideals and
3456:
is a proper ideal. More generally, for each left ideal
13727:. Cambridge: Cambridge University Press. p. 132.
12925:
An ideal can also be thought of as a specific type of
7936:{\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}}}
6838:
6778:
6605:
4292:
1612:(often without the multiplicative identity) such as a
749:
is derived from the notion of ideal in ring theory. A
13399:
Some authors call the zero and unit ideals of a ring
13318:
13286:
13259:
13235:
13211:
13181:
13154:
13132:
13112:
13090:
13068:
13048:
13028:
13006:
12980:
12974:-module (by left multiplication), then a left ideal
12960:
12940:
12904:
12854:
12806:
12762:
12734:
12694:
12654:
12632:
12608:
12581:
12559:
12537:
12514:
12482:
12448:
12409:
12383:
12338:
12300:
12268:
12238:
12206:
12171:
12139:
12062:
12016:
11990:
11945:
11840:
11821:
11790:
11753:
11734:
11703:
11655:
11609:
11569:
11497:
11412:
11327:
11274:
11237:
11195:
11142:
11103:
11060:
11020:
10992:
10945:
10900:
10863:
10844:
10817:
10786:
10758:
10715:
10684:
10634:
10606:
10481:
10443:
10409:
10378:
10322:
10294:
10190:
10119:
10070:
10020:
9968:
9934:
9883:
9793:
9758:
9723:
9672:
9624:
9596:
9555:
9513:
9480:
9406:
9369:
9339:
9315:
9283:
9211:
9182:
9154:
9099:
9054:
8914:
8863:
8763:
8555:
8496:
8390:
8269:
8212:
8187:
8165:
8127:
8057:
8032:
7975:
7949:
7915:
7787:
7751:
7721:
7674:
7620:
7582:
7534:
7510:
7486:
7393:
7303:
7216:
7165:
7134:
7108:
7077:
7041:
7015:
6991:
6957:
6928:
6898:
6684:
6647:
6555:
6522:
6496:
6336:
6252:
6218:
6142:
6114:
6082:
5869:
5847:
5827:
5805:
5780:
5758:
5736:
5716:
5692:
5664:
5632:
5541:
5519:
5467:
5443:
5421:
5399:
5377:
5343:
5315:
5293:
5271:
5210:
5182:
5124:
5096:
5068:
5031:
5006:
4813:
4661:
4620:
4591:
4532:
4487:
4461:
4422:
4381:
4355:
4291:
4263:
4213:
4193:
4139:
4090:
4064:
4040:
3980:
3938:
3850:
3821:
3759:
3722:
3702:
3682:
3662:
3628:
3583:
3546:
3517:
3470:
3430:
3388:
3366:
3315:
3291:
3225:
3189:
3142:
3099:
3062:
3036:
3016:
2985:
2948:
2926:
2900:
2878:
2858:
2824:
2797:
2771:
2737:
2713:
2685:
2649:
2629:
2609:
2580:
2546:
2522:
2500:
2464:
2429:
2396:
2370:
2343:
2321:
2294:
2270:
2201:
2181:
2152:
2130:
2099:
2064:
2041:
2021:
1983:
1941:
1905:
1845:
1815:
1791:
1752:
1707:
1678:
1650:
1585:
1547:
1521:
1483:
1457:
1430:
1352:
1321:
1264:
1217:
1188:
1135:
1104:
1059:
1034:
1006:
978:
941:
903:
880:
858:
836:
726:
can be generalized to ideals. There is a version of
506:
468:
431:
211:
184:
13899:
Dummit, David Steven; Foote, Richard Martin (2004).
12368:{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}}
12047:{\displaystyle {\mathfrak {p}}={\mathfrak {p}}^{ec}}
11686:{\displaystyle {\mathfrak {b}}^{ce}={\mathfrak {b}}}
11640:{\displaystyle {\mathfrak {a}}^{ec}={\mathfrak {a}}}
9713:
admits a maximal submodule, in particular, one has:
2035:
is a nonzero element, then the principal left ideal
13753:
13539:
10113:is an ideal properly minimal over the latter, then
7707:{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(1)}
7062:{\displaystyle {\mathfrak {a}}\cup {\mathfrak {b}}}
13326:
13300:
13270:
13243:
13219:
13195:
13160:
13138:
13118:
13096:
13074:
13054:
13034:
13012:
12986:
12966:
12946:
12910:
12884:
12836:
12780:
12746:
12718:
12678:
12638:
12614:
12587:
12565:
12543:
12520:
12488:
12466:
12419:
12393:
12367:
12322:
12284:
12248:
12222:
12186:
12155:
12123:
12046:
12000:
11977:under extension is one of the central problems of
11965:
11857:
11827:
11800:
11770:
11740:
11713:
11685:
11639:
11591:
11541:
11479:
11394:
11311:
11256:
11219:
11181:
11128:
11087:
11037:
11002:
10975:
10930:
10880:
10850:
10827:
10796:
10768:
10725:
10701:
10663:
10616:
10589:
10462:
10419:
10395:
10341:
10304:
10242:
10174:
10105:
10052:
10004:
9952:
9901:
9858:
9770:
9738:
9699:
9656:
9608:
9570:
9528:
9495:
9463:
9387:
9349:
9325:
9301:
9269:
9197:
9160:
9137:
9078:
8985:
8900:
8848:
8748:
8540:
8482:
8371:
8253:
8193:
8171:
8159:is the set of integers that are divisible by both
8151:
8110:
8040:
8004:
7959:
7935:
7892:
7761:
7731:
7706:
7657:
7602:
7544:
7520:
7496:
7469:
7370:
7283:
7195:
7144:
7118:
7094:
7061:
7025:
7001:
6977:
6938:
6908:
6873:
6667:
6629:
6532:
6506:
6342:
6304:
6224:
6154:
6126:
6100:
5875:
5853:
5833:
5811:
5789:
5764:
5742:
5722:
5698:
5670:
5644:
5547:
5525:
5503:
5449:
5427:
5405:
5383:
5361:
5327:
5299:
5277:
5248:
5194:
5168:
5108:
5080:
5052:
5015:
4978:
4783:
4629:
4600:
4550:
4516:
4473:
4439:
4404:
4367:
4326:
4275:
4247:
4199:
4168:
4116:
4074:
4050:
4018:
3958:
3920:
3833:
3790:
3743:
3708:
3688:
3668:
3646:
3598:
3561:
3532:
3495:
3448:
3414:
3372:
3350:
3297:
3275:
3207:
3166:
3117:
3083:
3048:
3022:
3002:
2969:
2932:
2908:
2886:
2864:
2841:
2803:
2783:
2755:
2719:
2697:
2671:
2635:
2615:
2592:
2564:
2528:
2506:
2481:
2448:
2404:
2378:
2354:
2327:
2305:
2278:
2249:
2187:
2167:
2136:
2114:
2085:
2050:
2027:
2007:
1965:
1917:
1889:
1831:
1801:
1764:
1726:
1691:
1662:
1597:
1571:
1533:
1495:
1469:
1439:
1372:
1335:
1278:
1235:
1200:
1182:is a left, right or two-sided ideal, the relation
1150:
1119:
1065:
1043:
1018:
990:
959:
921:
886:
864:
842:
530:
483:
446:
238:
192:
13657:Because simple commutative rings are fields. See
13524:
12008:is a contraction of a prime ideal if and only if
11984:The following is sometimes useful: a prime ideal
10582:
10501:
9869:where the intersection on the left is called the
4117:{\displaystyle ({\mathfrak {a}}:{\mathfrak {b}})}
14077:
12323:{\displaystyle {\mathfrak {q}}B_{\mathfrak {p}}}
9784:A maximal ideal is a prime ideal and so one has
7157:The distributive law holds for two-sided ideals
4801:.) A right (resp. two-sided) ideal generated by
3696:and the left (resp. right, two-sided) ideals of
2015:are the only left (or right) ideals. (Proof: if
13870:
13858:
11966:{\displaystyle {\mathfrak {a}}={\mathfrak {p}}}
11592:{\displaystyle {\mathfrak {a}}\supseteq \ker f}
8021:
7603:{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}
6978:{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}}
6668:{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}
5658:: the whole ring (being the ideal generated by
3959:{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}
2920:contains the ideal of all continuous functions
1890:{\displaystyle r=(ru^{-1})u\in {\mathfrak {a}}}
13509:
13376:Splitting of prime ideals in Galois extensions
12599:if it satisfies the following two conditions:
11268:(and therefore not maximal, as well). Indeed,
7095:{\displaystyle {\mathfrak {a}}{\mathfrak {b}}}
4575:, there is the smallest left ideal containing
2815:is zero is a left ideal but not a right ideal.
2729:. It is not a left ideal. Similarly, for each
1424:is a subrng with the additional property that
6641:which is a left (resp. right) ideal, and, if
6413:. Despite their names, fractional ideals are
3804:section for the definitions of these ideals).
2643:-th row is zero is a right ideal in the ring
635:
12165:, a contradiction. Now, the prime ideals of
9455:
9413:
6865:
6702:
6624:
6576:
5639:
5633:
5498:
5474:
5163:
5131:
4969:
4826:
4775:
4671:
4545:
4539:
4405:{\displaystyle {\mathfrak {a}}_{0}\subset R}
3915:
3876:
3106:
3100:
2364:. In fact, every non-zero ideal of the ring
1721:
1708:
1164:is a left ideal that is also a right ideal.
239:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
13898:
13705:
13693:
13645:
13621:
13602:
13590:
13578:
13566:
13554:
13533:
13505:
13503:
5604:. The factor ring of a maximal ideal is a
13711:
12885:{\displaystyle x\otimes r\in (I,\otimes )}
12837:{\displaystyle r\otimes x\in (I,\otimes )}
5945:. The factor ring of a radical ideal is a
5415:. Conversely, given a congruence relation
4169:{\displaystyle {\mathfrak {a}}_{i},i\in S}
1734:consisting of only the additive identity 0
1619:
642:
628:
13717:
13320:
13294:
13264:
13237:
13213:
13189:
11111:
11070:
11062:
9700:{\displaystyle M=JM\subset L\subsetneq M}
9539:The following simple but important fact (
9440:
9079:{\displaystyle J=\operatorname {Jac} (R)}
8220:
8104:
8034:
4975:
4908:
4733:
4451:. (Again this is still valid if the ring
2993:
2902:
2880:
2832:
2466:
2398:
2372:
2348:
2299:
2272:
1258:forms a left, right or bi module denoted
1098:is defined similarly, with the condition
741:The related, but distinct, concept of an
531:{\displaystyle \mathbb {Z} (p^{\infty })}
508:
471:
434:
232:
219:
186:
14060:
13917:
13747:
13500:
13227:is an abelian group that is a subset of
9588:, since if there is a maximal submodule
9029:be a commutative ring. By definition, a
5885:. The factor ring of a prime ideal is a
5169:{\displaystyle X=\{x_{1},\dots ,x_{n}\}}
4338:. (Note: this fact remains true even if
11480:{\displaystyle (1-i)=((1+i)-(1+i)^{2})}
11395:{\displaystyle (1+i)=((1-i)-(1-i)^{2})}
10664:{\displaystyle f^{-1}({\mathfrak {b}})}
9953:{\displaystyle \operatorname {Jac} (R)}
9902:{\displaystyle \operatorname {nil} (R)}
9443: is a unit element for every
9302:{\displaystyle \operatorname {Ann} (M)}
7665:in the following two cases (at least)
4517:{\displaystyle {\mathfrak {a}}_{0}=(0)}
3620:: Given a surjective ring homomorphism
1079:In other words, a left ideal is a left
14:
14078:
14010:
13802:"sums, products, and powers of ideals"
13759:
13662:A First Course in Noncommutative Rings
6542:, left (resp. right) ideals of a ring
5592:if there exists no other proper ideal
4412:is a left ideal that is disjoint from
3612:#Extension and contraction of an ideal
10256:Extension and contraction of an ideal
10014:. (Proof: first note the DCC implies
7739:is generated by elements that form a
6373:. (This is stronger than saying that
4579:, called the left ideal generated by
3276:{\displaystyle \ker(f)=f^{-1}(0_{S})}
1404:
13962:
13633:
13126:is a two-sided ideal if it is a sub-
13062:if and only if it is a left (right)
12056:. (Proof: Assuming the latter, note
9004:
7565:. The lattice is not, in general, a
7102:is contained in the intersection of
5504:{\displaystyle I=\{x\in R:x\sim 0\}}
5249:{\displaystyle (x_{1},\dots ,x_{n})}
3801:
1832:{\displaystyle u\in {\mathfrak {a}}}
99:Free product of associative algebras
13929:, vol. 150, Berlin, New York:
13881:Introduction to Commutative Algebra
13658:
13310:. So these give all the ideals of
12412:
12386:
12359:
12342:
12314:
12303:
12277:
12241:
12215:
12178:
12148:
12110:
12098:
12083:
12066:
12030:
12019:
11993:
11958:
11948:
11858:{\displaystyle {\mathfrak {a}}^{e}}
11844:
11793:
11771:{\displaystyle {\mathfrak {a}}^{e}}
11757:
11706:
11678:
11659:
11632:
11613:
11572:
11312:{\displaystyle (1\pm i)^{2}=\pm 2i}
11038:{\displaystyle {\mathfrak {a}}^{e}}
11024:
10995:
10968:
10949:
10923:
10904:
10881:{\displaystyle {\mathfrak {b}}^{c}}
10867:
10820:
10789:
10761:
10718:
10702:{\displaystyle {\mathfrak {b}}^{c}}
10688:
10653:
10609:
10556:
10485:
10452:
10412:
10396:{\displaystyle {\mathfrak {a}}^{e}}
10382:
10331:
10297:
10229:
10203:
10131:
10073:
9833:
9819:
9709:, a contradiction. Since a nonzero
9380:
9342:
9318:
9309:is a maximal ideal. Conversely, if
9127:
9113:
8978:
8971:
8927:
8917:
8893:
8886:
8876:
8866:
8773:
8766:
8565:
8558:
8509:
8499:
8403:
8393:
8340:
8300:
8272:
7997:
7990:
7979:
7952:
7928:
7918:
7885:
7878:
7863:
7853:
7837:
7819:
7754:
7724:
7687:
7677:
7650:
7643:
7633:
7623:
7595:
7585:
7537:
7513:
7489:
7462:
7452:
7442:
7432:
7419:
7409:
7396:
7363:
7356:
7346:
7339:
7329:
7319:
7309:
7276:
7269:
7259:
7252:
7239:
7229:
7219:
7188:
7178:
7168:
7137:
7111:
7087:
7080:
7054:
7044:
7018:
6994:
6970:
6960:
6931:
6901:
6799:
6772:
6694:
6687:
6660:
6650:
6619:
6599:
6568:
6558:
6525:
6499:
6485:
4491:
4440:{\displaystyle {\mathfrak {a}}_{0}}
4426:
4385:
4312:
4234:
4217:
4143:
4106:
4096:
4067:
4043:
4011:
3996:
3986:
3951:
3941:
3180:over a skew-field is a simple ring.
1882:
1824:
1794:
1740:forms a two-sided ideal called the
1383:that associates to each element of
24:
14016:Introduction to algebraic K-theory
13518:
12432:
10463:{\displaystyle f({\mathfrak {a}})}
10342:{\displaystyle f({\mathfrak {a}})}
9037:is the annihilator of a (nonzero)
8997:computations can be checked using
5561:
3791:{\displaystyle J\mapsto f^{-1}(J)}
3118:{\displaystyle \vert x\vert >L}
1636:itself forms a two-sided ideal of
520:
25:
14112:
14054:
13301:{\displaystyle m\in \mathbb {Z} }
12719:{\displaystyle x\in (I,\otimes )}
12679:{\displaystyle r\in (R,\otimes )}
12437:Ideals can be generalized to any
12285:{\displaystyle A-{\mathfrak {p}}}
12223:{\displaystyle A-{\mathfrak {p}}}
12187:{\displaystyle B_{\mathfrak {p}}}
12156:{\displaystyle A-{\mathfrak {p}}}
11542:{\displaystyle (2)^{e}=(1+i)^{2}}
9388:{\displaystyle R/{\mathfrak {m}}}
9357:is the annihilator of the simple
8016:fundamental theorem of arithmetic
7610:are ideals of a commutative ring
6244:of the associated quotient ring,
5062:). The principal two-sided ideal
3716:: the correspondence is given by
709:: in this ring, every ideal is a
587:Noncommutative algebraic geometry
14061:Levinson, Jake (July 14, 2014).
13542:An introduction to number theory
13432:, we must allow the addition of
12232:. Hence, there is a prime ideal
11828:{\displaystyle \Leftrightarrow }
11741:{\displaystyle \Leftrightarrow }
11189:where (one can show) neither of
6358:is equal to the height of every
6038:: A left primitive ideal is the
3807:(For those who know modules) If
3030:for which there exists a number
2058:(see below) is nonzero and thus
484:{\displaystyle \mathbb {Q} _{p}}
447:{\displaystyle \mathbb {Z} _{p}}
13823:
13794:
13765:
13741:
13669:
13651:
13627:
13414:
12798:is defined with the condition "
12420:{\displaystyle {\mathfrak {p}}}
12394:{\displaystyle {\mathfrak {q}}}
12249:{\displaystyle {\mathfrak {q}}}
12001:{\displaystyle {\mathfrak {p}}}
11801:{\displaystyle {\mathfrak {a}}}
11714:{\displaystyle {\mathfrak {a}}}
11003:{\displaystyle {\mathfrak {a}}}
10828:{\displaystyle {\mathfrak {b}}}
10797:{\displaystyle {\mathfrak {b}}}
10769:{\displaystyle {\mathfrak {a}}}
10726:{\displaystyle {\mathfrak {b}}}
10617:{\displaystyle {\mathfrak {b}}}
10420:{\displaystyle {\mathfrak {a}}}
10305:{\displaystyle {\mathfrak {a}}}
9350:{\displaystyle {\mathfrak {m}}}
9326:{\displaystyle {\mathfrak {m}}}
7960:{\displaystyle {\mathfrak {c}}}
7905:An integral domain is called a
7762:{\displaystyle {\mathfrak {b}}}
7732:{\displaystyle {\mathfrak {a}}}
7545:{\displaystyle {\mathfrak {c}}}
7521:{\displaystyle {\mathfrak {b}}}
7497:{\displaystyle {\mathfrak {a}}}
7145:{\displaystyle {\mathfrak {b}}}
7119:{\displaystyle {\mathfrak {a}}}
7026:{\displaystyle {\mathfrak {b}}}
7002:{\displaystyle {\mathfrak {a}}}
6939:{\displaystyle {\mathfrak {b}}}
6909:{\displaystyle {\mathfrak {a}}}
6533:{\displaystyle {\mathfrak {b}}}
6507:{\displaystyle {\mathfrak {a}}}
6425:, then it is truly an ideal of
6402:: This is usually defined when
4375:is a possibly empty subset and
4075:{\displaystyle {\mathfrak {b}}}
4051:{\displaystyle {\mathfrak {a}}}
3540:is a left ideal of the subring
3415:{\displaystyle 1_{S}\neq 0_{S}}
3003:{\displaystyle C(\mathbb {R} )}
2842:{\displaystyle C(\mathbb {R} )}
2250:{\displaystyle z=z(yx)=(zy)x=x}
1802:{\displaystyle {\mathfrak {a}}}
13596:
13584:
13572:
13560:
13548:
13393:
13196:{\displaystyle R=\mathbb {Z} }
12879:
12867:
12831:
12819:
12775:
12763:
12713:
12701:
12673:
12661:
12461:
12449:
12330:is a maximal ideal containing
12104:
11822:
11735:
11530:
11517:
11505:
11498:
11474:
11465:
11452:
11446:
11434:
11431:
11425:
11413:
11389:
11380:
11367:
11361:
11349:
11346:
11340:
11328:
11288:
11275:
11245:
11238:
11176:
11164:
11161:
11149:
11066:
10986:It is false, in general, that
10845:
10658:
10648:
10535:
10522:
10457:
10447:
10431:is defined to be the ideal in
10336:
10326:
10163:
10160:
10147:
10126:
10100:
10087:
9999:
9993:
9981:
9975:
9947:
9941:
9896:
9890:
9853:
9847:
9806:
9800:
9657:{\displaystyle J\cdot (M/L)=0}
9645:
9631:
9296:
9290:
9258:
9252:
9232:
9226:
9073:
9067:
8963:
8935:
8843:
8781:
8743:
8663:
8657:
8654:
8642:
8633:
8621:
8612:
8600:
8591:
8579:
8573:
8535:
8517:
8477:
8453:
8447:
8411:
8366:
8348:
8332:
8308:
8292:
8280:
8254:{\displaystyle R=\mathbb {C} }
8248:
8224:
8146:
8140:
8134:
8128:
8100:
8088:
8076:
8070:
8064:
8058:
7868:
7848:
7842:
7806:
7701:
7695:
7424:
7404:
7324:
7304:
7244:
7224:
6437:: Usually an invertible ideal
6299:
6285:
6266:
6260:
5243:
5211:
5103:
5097:
4511:
4505:
4111:
4091:
4001:
3981:
3932:is a left ideal. Given ideals
3870:
3864:
3785:
3779:
3763:
3738:
3732:
3726:
3638:
3593:
3587:
3556:
3550:
3527:
3521:
3490:
3484:
3443:
3437:
3351:{\displaystyle f(1_{R})=1_{S}}
3332:
3319:
3270:
3257:
3238:
3232:
3199:
3161:
3155:
3149:
3143:
3072:
3066:
2997:
2989:
2958:
2952:
2836:
2828:
2666:
2660:
2476:
2470:
2235:
2226:
2220:
2211:
2080:
2074:
2002:
1996:
1990:
1984:
1960:
1954:
1948:
1942:
1871:
1852:
1759:
1753:
1657:
1651:
1644:. It is often also denoted by
1356:
954:
942:
916:
904:
798:
784:Vorlesungen ΓΌber Zahlentheorie
525:
512:
13:
1:
13927:Graduate Texts in Mathematics
13859:Atiyah & Macdonald (1969)
13493:
13482:in the natural numbers. When
13271:{\displaystyle m\mathbb {Z} }
10053:{\displaystyle J^{n}=J^{n+1}}
9609:{\displaystyle L\subsetneq M}
6406:is a commutative domain with
6354:(in height) if the height of
4345:The above fact together with
4207:is a totally ordered set and
3744:{\displaystyle I\mapsto f(I)}
2756:{\displaystyle 1\leq j\leq n}
2565:{\displaystyle 1\leq i\leq n}
2355:{\displaystyle n\mathbb {Z} }
2306:{\displaystyle 2\mathbb {Z} }
1343:is a ring, and the function
734:(a type of ring important in
13540:Everest G., Ward T. (2005).
13327:{\displaystyle \mathbb {Z} }
13244:{\displaystyle \mathbb {Z} }
13220:{\displaystyle \mathbb {Z} }
13082:-module that is a subset of
12781:{\displaystyle (I,\otimes )}
12467:{\displaystyle (R,\otimes )}
12429:. The converse is obvious.)
11182:{\displaystyle 2=(1+i)(1-i)}
11010:being prime (or maximal) in
10851:{\displaystyle \Rightarrow }
10365:into the field of rationals
9750:is finitely generated, then
8041:{\displaystyle \mathbb {Z} }
8022:Examples of ideal operations
7480:where the equality holds if
6481:in rings other than domains.
6188:: Some power of it is zero.
5949:for general rings, and is a
5391:is a congruence relation on
2909:{\displaystyle \mathbb {R} }
2887:{\displaystyle \mathbb {R} }
2482:{\displaystyle \mathbb {R} }
2405:{\displaystyle \mathbb {Z} }
2379:{\displaystyle \mathbb {Z} }
2279:{\displaystyle \mathbb {Z} }
690:in a way similar to how, in
193:{\displaystyle \mathbb {Z} }
7:
13994:Algebras, rings and modules
13512:Mathematics and its history
13356:Boolean prime ideal theorem
13351:Noether isomorphism theorem
13339:
13042:is a left (right) ideal of
11136:, the element 2 factors as
8152:{\displaystyle (n)\cap (m)}
7909:if for each pair of ideals
1387:its equivalence class is a
1304:and is a generalization of
718:of a ring are analogous to
698:can be used to construct a
658:, and more specifically in
345:Unique factorization domain
10:
14117:
14020:Princeton University Press
13525:Harold M. Edwards (1977).
12747:{\displaystyle r\otimes x}
9008:
6196:: an ideal generated by a
4368:{\displaystyle E\subset R}
3834:{\displaystyle S\subset M}
2264:form an ideal in the ring
1572:{\displaystyle r=r1\in I;}
1515:were a subring, for every
1394:that has the ideal as its
1300:. (It is an instance of a
760:
728:unique prime factorization
105:Tensor product of algebras
13996:. Vol. 1. Springer.
13939:10.1007/978-1-4612-5350-1
13706:Dummit & Foote (2004)
13694:Dummit & Foote (2004)
13646:Dummit & Foote (2004)
13622:Dummit & Foote (2004)
13581:, Β§ 10.1., Proposition 3.
11045:is prime (or maximal) in
9711:finitely generated module
9333:is a maximal ideal, then
5088:is often also denoted by
4455:lacks the unity 1.) When
4349:proves the following: if
3676:containing the kernel of
3496:{\displaystyle f^{-1}(I)}
3380:is not the zero ring (so
2784:{\displaystyle n\times n}
2705:matrices with entries in
2698:{\displaystyle n\times n}
2593:{\displaystyle n\times n}
1727:{\displaystyle \{0_{R}\}}
724:Chinese remainder theorem
13831:"intersection of ideals"
13569:, Β§ 10.1., Examples (1).
13386:
12496:is the object where the
10752:is a ring homomorphism,
10361:of the ring of integers
10349:need not be an ideal in
9826: prime ideals
9172:is a nonzero element in
6026:: This type of ideal is
6024:Finitely generated ideal
6016:: An ideal generated by
5362:{\displaystyle x-y\in I}
4342:is without the unity 1.)
3647:{\displaystyle f:R\to S}
3606:need not be an ideal of
3283:is a two-sided ideal of
3208:{\displaystyle f:R\to S}
2918:pointwise multiplication
2672:{\displaystyle M_{n}(R)}
1839:is a unit element, then
1373:{\displaystyle R\to R/I}
1236:{\displaystyle x-y\in I}
768:invented the concept of
383:Formal power series ring
333:Integrally closed domain
14101:Algebraic number theory
13724:Commutative Ring Theory
13603:Dummit & Foote 2004
13593:, Ch. 7, Proposition 6.
13591:Dummit & Foote 2004
13579:Dummit & Foote 2004
13567:Dummit & Foote 2004
13555:Dummit & Foote 2004
13510:John Stillwell (2010).
13454:-fold sums of the form
13436:-fold sums of the form
12377:. One then checks that
12198:that are disjoint from
12194:correspond to those in
11979:algebraic number theory
11257:{\displaystyle (2)^{e}}
11220:{\displaystyle 1+i,1-i}
9168:is a simple module and
5989:, then at least one of
5821:, then at least one of
5328:{\displaystyle x\sim y}
4551:{\displaystyle E=\{1\}}
4474:{\displaystyle R\neq 0}
4130:in commutative algebra.
4126:; it is an instance of
3449:{\displaystyle \ker(f)}
3167:{\displaystyle (0),(1)}
2449:{\displaystyle x^{2}+1}
2008:{\displaystyle (0),(1)}
1966:{\displaystyle (0),(1)}
1620:Examples and properties
1201:{\displaystyle x\sim y}
1151:{\displaystyle xr\in I}
1120:{\displaystyle rx\in I}
392:Algebraic number theory
85:Total ring of fractions
13328:
13302:
13272:
13245:
13221:
13197:
13162:
13140:
13120:
13098:
13076:
13056:
13036:
13014:
12994:is really just a left
12988:
12968:
12948:
12912:
12886:
12838:
12782:
12748:
12720:
12680:
12640:
12616:
12589:
12567:
12545:
12522:
12490:
12468:
12421:
12395:
12369:
12324:
12286:
12250:
12224:
12188:
12157:
12125:
12048:
12002:
11967:
11865:is a maximal ideal in
11859:
11829:
11802:
11772:
11742:
11715:
11687:
11641:
11593:
11554:On the other hand, if
11543:
11481:
11396:
11313:
11258:
11221:
11183:
11130:
11089:
11039:
11004:
10977:
10932:
10882:
10852:
10829:
10798:
10770:
10727:
10703:
10671:is always an ideal of
10665:
10618:
10591:
10464:
10421:
10397:
10343:
10306:
10244:
10176:
10107:
10054:
10006:
9954:
9903:
9860:
9772:
9740:
9701:
9658:
9610:
9572:
9547:is a module such that
9530:
9497:
9465:
9389:
9351:
9327:
9303:
9271:
9199:
9162:
9139:
9080:
8987:
8902:
8850:
8750:
8542:
8484:
8373:
8255:
8195:
8173:
8153:
8112:
8042:
8006:
7961:
7937:
7894:
7763:
7733:
7708:
7659:
7604:
7546:
7522:
7498:
7471:
7372:
7285:
7197:
7146:
7120:
7096:
7071:), while the product
7063:
7027:
7003:
6979:
6940:
6910:
6875:
6669:
6631:
6534:
6508:
6344:
6306:
6226:
6156:
6155:{\displaystyle y\in J}
6128:
6127:{\displaystyle x\in I}
6102:
5953:for commutative rings.
5893:for commutative rings.
5877:
5855:
5835:
5813:
5791:
5766:
5744:
5724:
5700:
5672:
5646:
5612:for commutative rings.
5549:
5527:
5505:
5451:
5429:
5407:
5385:
5363:
5329:
5301:
5279:
5250:
5196:
5176:is a finite set, then
5170:
5110:
5082:
5054:
5053:{\displaystyle xR,RxR}
5017:
4980:
4785:
4637:is the set of all the
4631:
4602:
4552:
4518:
4475:
4441:
4406:
4369:
4328:
4277:
4276:{\displaystyle i<j}
4249:
4201:
4170:
4118:
4076:
4052:
4020:
3966:of a commutative ring
3960:
3922:
3835:
3792:
3745:
3710:
3690:
3670:
3648:
3600:
3563:
3534:
3497:
3450:
3416:
3374:
3352:
3299:
3277:
3209:
3168:
3119:
3085:
3084:{\displaystyle f(x)=0}
3050:
3049:{\displaystyle L>0}
3024:
3004:
2971:
2970:{\displaystyle f(1)=0}
2934:
2910:
2888:
2866:
2843:
2805:
2785:
2757:
2721:
2699:
2673:
2637:
2617:
2594:
2566:
2530:
2508:
2483:
2450:
2414:principal ideal domain
2406:
2380:
2356:
2329:
2307:
2280:
2251:
2189:
2169:
2138:
2116:
2087:
2086:{\displaystyle Rx=(1)}
2052:
2029:
2009:
1967:
1919:
1918:{\displaystyle r\in R}
1891:
1833:
1803:
1785:). Note: a left ideal
1766:
1728:
1693:
1664:
1599:
1573:
1535:
1534:{\displaystyle r\in R}
1497:
1496:{\displaystyle x\in I}
1471:
1470:{\displaystyle r\in R}
1441:
1374:
1337:
1280:
1237:
1202:
1152:
1121:
1067:
1045:
1020:
1019:{\displaystyle x\in I}
992:
991:{\displaystyle r\in R}
961:
923:
888:
866:
844:
549:Noncommutative algebra
532:
485:
448:
400:Algebraic number field
351:Principal ideal domain
240:
194:
132:Frobenius endomorphism
13968:Undergraduate Algebra
13329:
13303:
13273:
13246:
13222:
13198:
13163:
13141:
13121:
13099:
13077:
13057:
13037:
13015:
12989:
12969:
12949:
12913:
12887:
12839:
12783:
12749:
12721:
12681:
12641:
12617:
12590:
12568:
12546:
12523:
12491:
12469:
12422:
12396:
12370:
12325:
12287:
12251:
12225:
12189:
12158:
12126:
12049:
12003:
11968:
11936:. The behaviour of a
11908:, respectively. Then
11860:
11830:
11803:
11773:
11743:
11716:
11688:
11642:
11594:
11544:
11482:
11397:
11314:
11259:
11222:
11184:
11131:
11090:
11040:
11005:
10978:
10933:
10883:
10853:
10830:
10799:
10771:
10728:
10704:
10666:
10619:
10592:
10465:
10422:
10398:
10344:
10307:
10245:
10177:
10108:
10055:
10007:
9955:
9904:
9861:
9773:
9741:
9702:
9659:
9611:
9573:
9531:
9498:
9466:
9390:
9352:
9328:
9304:
9272:
9200:
9163:
9140:
9081:
8988:
8903:
8851:
8751:
8543:
8485:
8374:
8256:
8196:
8174:
8154:
8113:
8043:
8007:
7962:
7938:
7895:
7764:
7734:
7709:
7660:
7605:
7547:
7523:
7499:
7472:
7373:
7286:
7198:
7147:
7121:
7097:
7064:
7028:
7004:
6980:
6941:
6911:
6876:
6670:
6632:
6535:
6509:
6345:
6330:in a Noetherian ring
6314:. A perfect ideal is
6307:
6227:
6212:in a Noetherian ring
6157:
6129:
6103:
6101:{\displaystyle x+y=1}
5889:in general and is an
5878:
5856:
5836:
5814:
5792:
5767:
5745:
5725:
5701:
5673:
5647:
5645:{\displaystyle \{0\}}
5550:
5528:
5506:
5452:
5430:
5428:{\displaystyle \sim }
5408:
5386:
5384:{\displaystyle \sim }
5364:
5330:
5302:
5280:
5251:
5197:
5171:
5111:
5083:
5055:
5018:
4981:
4786:
4632:
4603:
4553:
4519:
4476:
4442:
4407:
4370:
4329:
4278:
4250:
4202:
4171:
4119:
4077:
4053:
4021:
3961:
3923:
3836:
3793:
3746:
3711:
3691:
3671:
3649:
3601:
3564:
3535:
3498:
3451:
3417:
3375:
3353:
3300:
3278:
3210:
3169:
3120:
3086:
3051:
3025:
3005:
2972:
2935:
2911:
2889:
2867:
2844:
2806:
2786:
2758:
2722:
2700:
2674:
2638:
2618:
2595:
2567:
2531:
2514:and positive integer
2509:
2484:
2451:
2407:
2381:
2357:
2330:
2308:
2281:
2252:
2190:
2170:
2139:
2117:
2088:
2053:
2030:
2010:
1968:
1920:
1892:
1834:
1804:
1767:
1729:
1694:
1692:{\displaystyle 1_{R}}
1665:
1610:non-associative rings
1600:
1574:
1536:
1498:
1472:
1442:
1375:
1338:
1281:
1238:
1203:
1153:
1122:
1068:
1046:
1021:
993:
962:
960:{\displaystyle (R,+)}
924:
922:{\displaystyle (I,+)}
889:
867:
845:
707:non-negative integers
533:
486:
449:
241:
195:
14091:Algebraic structures
14086:Ideals (ring theory)
14012:Milnor, John Willard
13366:Ideal (order theory)
13316:
13284:
13257:
13233:
13209:
13179:
13152:
13130:
13110:
13088:
13066:
13046:
13026:
13004:
12978:
12958:
12938:
12902:
12852:
12804:
12760:
12732:
12692:
12652:
12630:
12606:
12579:
12557:
12535:
12512:
12480:
12446:
12407:
12381:
12336:
12298:
12266:
12236:
12204:
12169:
12137:
12060:
12014:
11988:
11943:
11838:
11819:
11788:
11778:is a prime ideal in
11751:
11732:
11701:
11653:
11607:
11567:
11495:
11410:
11325:
11272:
11235:
11193:
11140:
11101:
11058:
11018:
10990:
10943:
10898:
10861:
10842:
10815:
10784:
10756:
10713:
10682:
10632:
10604:
10479:
10441:
10407:
10376:
10320:
10292:
10252:, a contradiction.)
10188:
10117:
10068:
10018:
9966:
9932:
9881:
9791:
9756:
9739:{\displaystyle JM=M}
9721:
9670:
9622:
9594:
9571:{\displaystyle JM=M}
9553:
9529:{\displaystyle 1-xy}
9511:
9496:{\displaystyle 1-yx}
9478:
9404:
9367:
9337:
9313:
9281:
9209:
9198:{\displaystyle Rx=M}
9180:
9152:
9120: maximal ideals
9097:
9052:
8912:
8861:
8761:
8553:
8494:
8388:
8267:
8210:
8185:
8163:
8125:
8055:
8030:
7973:
7947:
7943:, there is an ideal
7913:
7785:
7749:
7719:
7672:
7618:
7580:
7567:distributive lattice
7532:
7508:
7484:
7391:
7301:
7214:
7163:
7132:
7106:
7075:
7039:
7013:
6989:
6955:
6926:
6896:
6682:
6645:
6553:
6520:
6494:
6387:equidimensional ring
6334:
6250:
6242:projective dimension
6216:
6198:system of parameters
6140:
6112:
6080:
5867:
5845:
5825:
5803:
5778:
5756:
5734:
5714:
5690:
5662:
5630:
5608:in general and is a
5539:
5517:
5465:
5441:
5419:
5397:
5375:
5341:
5313:
5291:
5269:
5263:congruence relations
5208:
5180:
5122:
5094:
5066:
5029:
5004:
4811:
4659:
4643:-linear combinations
4618:
4589:
4530:
4485:
4459:
4420:
4379:
4353:
4289:
4261:
4211:
4191:
4137:
4088:
4062:
4038:
3978:
3936:
3848:
3819:
3757:
3720:
3700:
3680:
3660:
3626:
3618:Ideal correspondence
3599:{\displaystyle f(I)}
3581:
3562:{\displaystyle f(R)}
3544:
3533:{\displaystyle f(I)}
3515:
3503:is a left ideal. If
3468:
3428:
3386:
3364:
3313:
3289:
3223:
3187:
3140:
3097:
3060:
3034:
3014:
2983:
2946:
2924:
2898:
2876:
2856:
2851:continuous functions
2822:
2795:
2769:
2735:
2711:
2683:
2647:
2627:
2607:
2578:
2544:
2520:
2498:
2462:
2427:
2394:
2368:
2341:
2335:is an ideal denoted
2319:
2292:
2268:
2199:
2179:
2168:{\displaystyle zy=1}
2150:
2128:
2115:{\displaystyle yx=1}
2097:
2062:
2039:
2019:
1981:
1939:
1903:
1843:
1813:
1789:
1750:
1705:
1676:
1648:
1583:
1545:
1519:
1481:
1455:
1428:
1350:
1319:
1262:
1248:equivalence relation
1215:
1186:
1133:
1102:
1057:
1032:
1004:
976:
939:
901:
878:
856:
834:
730:for the ideals of a
555:Noncommutative rings
504:
466:
429:
273:Non-associative ring
209:
182:
139:Algebraic structures
14096:Commutative algebra
13990:Hazewinkel, Michiel
13861:, Proposition 3.16.
13719:Matsumura, Hideyuki
13605:, Ch. 7, Theorem 7.
13173:Example: If we let
12500:structure has been
9909:is also the set of
9877:. As it turns out,
9771:{\displaystyle M=0}
7802:
5600:a proper subset of
5202:is also written as
5195:{\displaystyle RXR}
5109:{\displaystyle (x)}
5081:{\displaystyle RxR}
4334:is a left ideal of
3841:a subset, then the
3507:is a left ideal of
3130:A ring is called a
2979:. Another ideal in
1977:is a skew-field if
1765:{\displaystyle (0)}
1663:{\displaystyle (1)}
1598:{\displaystyle I=R}
1336:{\displaystyle R/I}
1302:congruence relation
1279:{\displaystyle R/I}
1256:equivalence classes
314:Commutative algebra
153:Associative algebra
35:Algebraic structure
13970:(Third ed.).
13872:Atiyah, Michael F.
13346:Modular arithmetic
13324:
13298:
13268:
13241:
13217:
13193:
13158:
13136:
13116:
13094:
13072:
13052:
13032:
13022:. In other words,
13010:
12984:
12964:
12944:
12934:. If we consider
12908:
12882:
12834:
12778:
12744:
12716:
12676:
12636:
12612:
12585:
12563:
12541:
12518:
12486:
12464:
12417:
12391:
12365:
12320:
12282:
12246:
12220:
12184:
12153:
12121:
12044:
11998:
11963:
11914:integral extension
11855:
11825:
11798:
11768:
11738:
11711:
11683:
11637:
11589:
11539:
11477:
11392:
11309:
11254:
11217:
11179:
11126:
11085:
11035:
11000:
10973:
10928:
10878:
10848:
10825:
10794:
10766:
10723:
10699:
10661:
10614:
10587:
10460:
10417:
10393:
10339:
10302:
10240:
10172:
10103:
10050:
10002:
9950:
9911:nilpotent elements
9899:
9856:
9830:
9768:
9736:
9697:
9654:
9606:
9568:
9526:
9493:
9461:
9385:
9347:
9323:
9299:
9267:
9195:
9158:
9135:
9124:
9076:
8983:
8898:
8846:
8746:
8538:
8480:
8369:
8251:
8191:
8169:
8149:
8108:
8038:
8002:
7957:
7933:
7890:
7788:
7759:
7729:
7704:
7655:
7600:
7542:
7518:
7494:
7467:
7368:
7281:
7193:
7142:
7116:
7092:
7059:
7023:
6999:
6975:
6936:
6906:
6871:
6842:
6782:
6665:
6627:
6609:
6530:
6504:
6340:
6302:
6222:
6152:
6124:
6098:
6028:finitely generated
5873:
5851:
5831:
5809:
5790:{\displaystyle ab}
5787:
5762:
5740:
5720:
5696:
5668:
5642:
5545:
5523:
5501:
5447:
5425:
5403:
5381:
5359:
5325:
5297:
5275:
5246:
5192:
5166:
5106:
5078:
5050:
5016:{\displaystyle Rx}
5013:
5000:and is denoted by
4976:
4781:
4630:{\displaystyle RX}
4627:
4601:{\displaystyle RX}
4598:
4583:and is denoted by
4548:
4514:
4471:
4447:and disjoint from
4437:
4402:
4365:
4324:
4323:
4308:
4273:
4245:
4197:
4166:
4114:
4082:and is denoted by
4072:
4048:
4016:
3956:
3918:
3831:
3788:
3751:and the pre-image
3741:
3706:
3686:
3666:
3644:
3596:
3559:
3530:
3493:
3446:
3412:
3370:
3348:
3295:
3273:
3219:, then the kernel
3205:
3164:
3115:
3081:
3046:
3020:
3000:
2967:
2930:
2906:
2884:
2862:
2839:
2801:
2781:
2753:
2717:
2695:
2669:
2633:
2613:
2590:
2562:
2526:
2504:
2479:
2446:
2402:
2388:Euclidean division
2376:
2352:
2325:
2303:
2276:
2247:
2185:
2165:
2134:
2112:
2083:
2051:{\displaystyle Rx}
2048:
2025:
2005:
1963:
1915:
1887:
1829:
1799:
1762:
1744:and is denoted by
1724:
1689:
1660:
1595:
1569:
1531:
1493:
1467:
1440:{\displaystyle rx}
1437:
1405:Note on convention
1370:
1333:
1306:modular arithmetic
1276:
1233:
1198:
1148:
1117:
1087:, considered as a
1063:
1044:{\displaystyle rx}
1041:
1016:
988:
957:
919:
884:
862:
840:
568:Semiprimitive ring
528:
481:
444:
252:Related structures
236:
190:
126:Inner automorphism
112:Ring homomorphisms
13981:978-0-387-22025-3
13948:978-0-387-94268-1
13884:. Perseus Books.
13876:Macdonald, Ian G.
13835:www.math.uiuc.edu
13806:www.math.uiuc.edu
13777:www.math.uiuc.edu
13750:, Exercise A 3.17
13161:{\displaystyle R}
13139:{\displaystyle R}
13119:{\displaystyle I}
13097:{\displaystyle R}
13075:{\displaystyle R}
13055:{\displaystyle R}
13035:{\displaystyle I}
13013:{\displaystyle R}
12987:{\displaystyle I}
12967:{\displaystyle R}
12947:{\displaystyle R}
12911:{\displaystyle R}
12639:{\displaystyle R}
12615:{\displaystyle I}
12588:{\displaystyle I}
12566:{\displaystyle R}
12544:{\displaystyle I}
12521:{\displaystyle R}
12489:{\displaystyle R}
11898:rings of integers
10286:ring homomorphism
10270:commutative rings
9960:is nilpotent and
9827:
9812:
9586:maximal submodule
9584:does not admit a
9444:
9161:{\displaystyle M}
9121:
9106:
9011:Radical of a ring
9005:Radical of a ring
8194:{\displaystyle m}
8172:{\displaystyle n}
6841:
6781:
6608:
6343:{\displaystyle R}
6326:: A proper ideal
6276:
6257:
6225:{\displaystyle R}
6208:: A proper ideal
6055:Irreducible ideal
5905:: A proper ideal
5876:{\displaystyle I}
5854:{\displaystyle b}
5834:{\displaystyle a}
5812:{\displaystyle I}
5765:{\displaystyle R}
5743:{\displaystyle b}
5723:{\displaystyle a}
5699:{\displaystyle I}
5686:: A proper ideal
5671:{\displaystyle 1}
5584:: A proper ideal
5548:{\displaystyle R}
5526:{\displaystyle I}
5450:{\displaystyle R}
5406:{\displaystyle R}
5300:{\displaystyle R}
5278:{\displaystyle I}
4293:
4285:. Then the union
4200:{\displaystyle S}
3709:{\displaystyle S}
3689:{\displaystyle f}
3669:{\displaystyle R}
3373:{\displaystyle S}
3307:. By definition,
3298:{\displaystyle R}
3217:ring homomorphism
3023:{\displaystyle f}
2933:{\displaystyle f}
2865:{\displaystyle f}
2804:{\displaystyle j}
2765:, the set of all
2720:{\displaystyle R}
2636:{\displaystyle i}
2616:{\displaystyle R}
2574:, the set of all
2529:{\displaystyle n}
2507:{\displaystyle R}
2328:{\displaystyle n}
2188:{\displaystyle z}
2175:for some nonzero
2137:{\displaystyle y}
2122:for some nonzero
2028:{\displaystyle x}
1392:ring homomorphism
1254:, and the set of
1066:{\displaystyle I}
887:{\displaystyle I}
865:{\displaystyle R}
843:{\displaystyle R}
652:
651:
609:Geometric algebra
320:Commutative rings
171:Category of rings
16:(Redirected from
14108:
14072:
14049:
14007:
13985:
13959:
13914:
13901:Abstract algebra
13895:
13862:
13856:
13850:
13849:
13847:
13846:
13837:. Archived from
13827:
13821:
13820:
13818:
13817:
13808:. Archived from
13798:
13792:
13791:
13789:
13788:
13779:. Archived from
13769:
13763:
13757:
13751:
13745:
13739:
13738:
13715:
13709:
13703:
13697:
13691:
13685:
13684:
13673:
13667:
13666:
13655:
13649:
13643:
13637:
13631:
13625:
13619:
13606:
13600:
13594:
13588:
13582:
13576:
13570:
13564:
13558:
13552:
13546:
13545:
13537:
13531:
13530:
13522:
13516:
13515:
13507:
13487:
13469:
13449:
13418:
13412:
13397:
13335:
13333:
13331:
13330:
13325:
13323:
13309:
13307:
13305:
13304:
13299:
13297:
13277:
13275:
13274:
13269:
13267:
13252:
13250:
13248:
13247:
13242:
13240:
13226:
13224:
13223:
13218:
13216:
13204:
13202:
13200:
13199:
13194:
13192:
13169:
13167:
13165:
13164:
13159:
13145:
13143:
13142:
13137:
13125:
13123:
13122:
13117:
13105:
13103:
13101:
13100:
13095:
13081:
13079:
13078:
13073:
13061:
13059:
13058:
13053:
13041:
13039:
13038:
13033:
13021:
13019:
13017:
13016:
13011:
12993:
12991:
12990:
12985:
12973:
12971:
12970:
12965:
12953:
12951:
12950:
12945:
12931:
12917:
12915:
12914:
12909:
12893:
12891:
12889:
12888:
12883:
12846:" replaced by "'
12845:
12843:
12841:
12840:
12835:
12789:
12787:
12785:
12784:
12779:
12753:
12751:
12750:
12745:
12727:
12725:
12723:
12722:
12717:
12685:
12683:
12682:
12677:
12645:
12643:
12642:
12637:
12621:
12619:
12618:
12613:
12594:
12592:
12591:
12586:
12574:
12572:
12570:
12569:
12564:
12550:
12548:
12547:
12542:
12527:
12525:
12524:
12519:
12495:
12493:
12492:
12487:
12475:
12473:
12471:
12470:
12465:
12428:
12426:
12424:
12423:
12418:
12416:
12415:
12400:
12398:
12397:
12392:
12390:
12389:
12376:
12374:
12372:
12371:
12366:
12364:
12363:
12362:
12352:
12351:
12346:
12345:
12329:
12327:
12326:
12321:
12319:
12318:
12317:
12307:
12306:
12293:
12291:
12289:
12288:
12283:
12281:
12280:
12260:, disjoint from
12255:
12253:
12252:
12247:
12245:
12244:
12231:
12229:
12227:
12226:
12221:
12219:
12218:
12193:
12191:
12190:
12185:
12183:
12182:
12181:
12164:
12162:
12160:
12159:
12154:
12152:
12151:
12130:
12128:
12127:
12122:
12120:
12119:
12114:
12113:
12103:
12102:
12101:
12088:
12087:
12086:
12076:
12075:
12070:
12069:
12055:
12053:
12051:
12050:
12045:
12043:
12042:
12034:
12033:
12023:
12022:
12007:
12005:
12004:
11999:
11997:
11996:
11972:
11970:
11969:
11964:
11962:
11961:
11952:
11951:
11864:
11862:
11861:
11856:
11854:
11853:
11848:
11847:
11834:
11832:
11831:
11826:
11807:
11805:
11804:
11799:
11797:
11796:
11777:
11775:
11774:
11769:
11767:
11766:
11761:
11760:
11747:
11745:
11744:
11739:
11720:
11718:
11717:
11712:
11710:
11709:
11694:
11692:
11690:
11689:
11684:
11682:
11681:
11672:
11671:
11663:
11662:
11646:
11644:
11643:
11638:
11636:
11635:
11626:
11625:
11617:
11616:
11598:
11596:
11595:
11590:
11576:
11575:
11550:
11548:
11546:
11545:
11540:
11538:
11537:
11513:
11512:
11489:, and therefore
11488:
11486:
11484:
11483:
11478:
11473:
11472:
11403:
11401:
11399:
11398:
11393:
11388:
11387:
11318:
11316:
11315:
11310:
11296:
11295:
11264:is not prime in
11263:
11261:
11260:
11255:
11253:
11252:
11226:
11224:
11223:
11218:
11188:
11186:
11185:
11180:
11135:
11133:
11132:
11127:
11125:
11114:
11096:
11094:
11092:
11091:
11086:
11084:
11073:
11065:
11044:
11042:
11041:
11036:
11034:
11033:
11028:
11027:
11009:
11007:
11006:
11001:
10999:
10998:
10982:
10980:
10979:
10974:
10972:
10971:
10962:
10961:
10953:
10952:
10937:
10935:
10934:
10929:
10927:
10926:
10917:
10916:
10908:
10907:
10887:
10885:
10884:
10879:
10877:
10876:
10871:
10870:
10857:
10855:
10854:
10849:
10834:
10832:
10831:
10826:
10824:
10823:
10803:
10801:
10800:
10795:
10793:
10792:
10775:
10773:
10772:
10767:
10765:
10764:
10732:
10730:
10729:
10724:
10722:
10721:
10708:
10706:
10705:
10700:
10698:
10697:
10692:
10691:
10670:
10668:
10667:
10662:
10657:
10656:
10647:
10646:
10623:
10621:
10620:
10615:
10613:
10612:
10596:
10594:
10593:
10588:
10586:
10585:
10573:
10572:
10560:
10559:
10550:
10549:
10534:
10533:
10518:
10517:
10505:
10504:
10495:
10494:
10489:
10488:
10471:
10469:
10467:
10466:
10461:
10456:
10455:
10426:
10424:
10423:
10418:
10416:
10415:
10402:
10400:
10399:
10394:
10392:
10391:
10386:
10385:
10348:
10346:
10345:
10340:
10335:
10334:
10311:
10309:
10308:
10303:
10301:
10300:
10251:
10249:
10247:
10246:
10241:
10233:
10232:
10226:
10225:
10207:
10206:
10200:
10199:
10181:
10179:
10178:
10173:
10159:
10158:
10140:
10135:
10134:
10112:
10110:
10109:
10104:
10099:
10098:
10077:
10076:
10059:
10057:
10056:
10051:
10049:
10048:
10030:
10029:
10013:
10011:
10009:
10008:
10003:
9959:
9957:
9956:
9951:
9908:
9906:
9905:
9900:
9865:
9863:
9862:
9857:
9837:
9836:
9829:
9828:
9825:
9823:
9822:
9779:
9777:
9775:
9774:
9769:
9745:
9743:
9742:
9737:
9708:
9706:
9704:
9703:
9698:
9663:
9661:
9660:
9655:
9641:
9617:
9615:
9613:
9612:
9607:
9579:
9577:
9575:
9574:
9569:
9541:Nakayama's lemma
9535:
9533:
9532:
9527:
9502:
9500:
9499:
9494:
9470:
9468:
9467:
9462:
9445:
9442:
9396:
9394:
9392:
9391:
9386:
9384:
9383:
9377:
9356:
9354:
9353:
9348:
9346:
9345:
9332:
9330:
9329:
9324:
9322:
9321:
9308:
9306:
9305:
9300:
9276:
9274:
9273:
9268:
9245:
9219:
9204:
9202:
9201:
9196:
9167:
9165:
9164:
9159:
9144:
9142:
9141:
9136:
9131:
9130:
9123:
9122:
9119:
9117:
9116:
9085:
9083:
9082:
9077:
9047:Jacobson radical
8992:
8990:
8989:
8984:
8982:
8981:
8975:
8974:
8962:
8961:
8931:
8930:
8921:
8920:
8907:
8905:
8904:
8899:
8897:
8896:
8890:
8889:
8880:
8879:
8870:
8869:
8855:
8853:
8852:
8847:
8842:
8841:
8802:
8801:
8777:
8776:
8770:
8769:
8755:
8753:
8752:
8747:
8742:
8741:
8675:
8674:
8569:
8568:
8562:
8561:
8547:
8545:
8544:
8539:
8513:
8512:
8503:
8502:
8489:
8487:
8486:
8481:
8407:
8406:
8397:
8396:
8380:
8378:
8376:
8375:
8370:
8344:
8343:
8304:
8303:
8276:
8275:
8260:
8258:
8257:
8252:
8223:
8202:
8200:
8198:
8197:
8192:
8178:
8176:
8175:
8170:
8158:
8156:
8155:
8150:
8117:
8115:
8114:
8109:
8107:
8047:
8045:
8044:
8039:
8037:
8013:
8011:
8009:
8008:
8003:
8001:
8000:
7994:
7993:
7984:
7983:
7982:
7966:
7964:
7963:
7958:
7956:
7955:
7942:
7940:
7939:
7934:
7932:
7931:
7922:
7921:
7901:
7899:
7897:
7896:
7891:
7889:
7888:
7882:
7881:
7875:
7867:
7866:
7857:
7856:
7841:
7840:
7834:
7823:
7822:
7816:
7801:
7796:
7770:
7768:
7766:
7765:
7760:
7758:
7757:
7741:regular sequence
7738:
7736:
7735:
7730:
7728:
7727:
7713:
7711:
7710:
7705:
7691:
7690:
7681:
7680:
7664:
7662:
7661:
7656:
7654:
7653:
7647:
7646:
7637:
7636:
7627:
7626:
7609:
7607:
7606:
7601:
7599:
7598:
7589:
7588:
7551:
7549:
7548:
7543:
7541:
7540:
7527:
7525:
7524:
7519:
7517:
7516:
7503:
7501:
7500:
7495:
7493:
7492:
7476:
7474:
7473:
7468:
7466:
7465:
7456:
7455:
7446:
7445:
7436:
7435:
7423:
7422:
7413:
7412:
7400:
7399:
7379:
7377:
7375:
7374:
7369:
7367:
7366:
7360:
7359:
7350:
7349:
7343:
7342:
7333:
7332:
7323:
7322:
7313:
7312:
7292:
7290:
7288:
7287:
7282:
7280:
7279:
7273:
7272:
7263:
7262:
7256:
7255:
7243:
7242:
7233:
7232:
7223:
7222:
7204:
7202:
7200:
7199:
7194:
7192:
7191:
7182:
7181:
7172:
7171:
7153:
7151:
7149:
7148:
7143:
7141:
7140:
7125:
7123:
7122:
7117:
7115:
7114:
7101:
7099:
7098:
7093:
7091:
7090:
7084:
7083:
7070:
7068:
7066:
7065:
7060:
7058:
7057:
7048:
7047:
7032:
7030:
7029:
7024:
7022:
7021:
7008:
7006:
7005:
7000:
6998:
6997:
6984:
6982:
6981:
6976:
6974:
6973:
6964:
6963:
6947:
6945:
6943:
6942:
6937:
6935:
6934:
6915:
6913:
6912:
6907:
6905:
6904:
6880:
6878:
6877:
6872:
6843:
6839:
6803:
6802:
6793:
6792:
6783:
6779:
6776:
6775:
6766:
6765:
6753:
6752:
6743:
6742:
6724:
6723:
6714:
6713:
6698:
6697:
6691:
6690:
6674:
6672:
6671:
6666:
6664:
6663:
6654:
6653:
6636:
6634:
6633:
6628:
6623:
6622:
6610:
6606:
6603:
6602:
6572:
6571:
6562:
6561:
6541:
6539:
6537:
6536:
6531:
6529:
6528:
6513:
6511:
6510:
6505:
6503:
6502:
6486:Ideal operations
6480:
6458:
6434:Invertible ideal
6399:Fractional ideal
6360:associated prime
6349:
6347:
6346:
6341:
6329:
6313:
6311:
6309:
6308:
6303:
6295:
6278:
6277:
6274:
6259:
6258:
6255:
6231:
6229:
6228:
6223:
6211:
6163:
6161:
6159:
6158:
6153:
6133:
6131:
6130:
6125:
6107:
6105:
6104:
6099:
6071:
6067:
6062:Comaximal ideals
6000:
5988:
5964:
5944:
5932:
5908:
5884:
5882:
5880:
5879:
5874:
5860:
5858:
5857:
5852:
5840:
5838:
5837:
5832:
5820:
5818:
5816:
5815:
5810:
5796:
5794:
5793:
5788:
5773:
5771:
5769:
5768:
5763:
5749:
5747:
5746:
5741:
5729:
5727:
5726:
5721:
5705:
5703:
5702:
5697:
5677:
5675:
5674:
5669:
5651:
5649:
5648:
5643:
5599:
5587:
5556:
5554:
5552:
5551:
5546:
5532:
5530:
5529:
5524:
5512:
5510:
5508:
5507:
5502:
5458:
5456:
5454:
5453:
5448:
5434:
5432:
5431:
5426:
5414:
5412:
5410:
5409:
5404:
5390:
5388:
5387:
5382:
5370:
5368:
5366:
5365:
5360:
5334:
5332:
5331:
5326:
5308:
5306:
5304:
5303:
5298:
5284:
5282:
5281:
5276:
5257:
5255:
5253:
5252:
5247:
5242:
5241:
5223:
5222:
5201:
5199:
5198:
5193:
5175:
5173:
5172:
5167:
5162:
5161:
5143:
5142:
5117:
5115:
5113:
5112:
5107:
5087:
5085:
5084:
5079:
5061:
5059:
5057:
5056:
5051:
5022:
5020:
5019:
5014:
4985:
4983:
4982:
4977:
4962:
4961:
4943:
4942:
4924:
4923:
4911:
4897:
4896:
4887:
4886:
4877:
4876:
4858:
4857:
4848:
4847:
4838:
4837:
4790:
4788:
4787:
4782:
4768:
4767:
4749:
4748:
4736:
4722:
4721:
4712:
4711:
4693:
4692:
4683:
4682:
4636:
4634:
4633:
4628:
4614:. Equivalently,
4609:
4607:
4605:
4604:
4599:
4559:
4557:
4555:
4554:
4549:
4523:
4521:
4520:
4515:
4501:
4500:
4495:
4494:
4480:
4478:
4477:
4472:
4446:
4444:
4443:
4438:
4436:
4435:
4430:
4429:
4411:
4409:
4408:
4403:
4395:
4394:
4389:
4388:
4374:
4372:
4371:
4366:
4333:
4331:
4330:
4325:
4322:
4321:
4316:
4315:
4307:
4284:
4282:
4280:
4279:
4274:
4254:
4252:
4251:
4246:
4244:
4243:
4238:
4237:
4227:
4226:
4221:
4220:
4206:
4204:
4203:
4198:
4175:
4173:
4172:
4167:
4153:
4152:
4147:
4146:
4125:
4123:
4121:
4120:
4115:
4110:
4109:
4100:
4099:
4081:
4079:
4078:
4073:
4071:
4070:
4057:
4055:
4054:
4049:
4047:
4046:
4025:
4023:
4022:
4017:
4015:
4014:
4008:
4000:
3999:
3990:
3989:
3974:-annihilator of
3965:
3963:
3962:
3957:
3955:
3954:
3945:
3944:
3927:
3925:
3924:
3919:
3860:
3859:
3840:
3838:
3837:
3832:
3799:
3797:
3795:
3794:
3789:
3778:
3777:
3750:
3748:
3747:
3742:
3715:
3713:
3712:
3707:
3695:
3693:
3692:
3687:
3675:
3673:
3672:
3667:
3655:
3653:
3651:
3650:
3645:
3605:
3603:
3602:
3597:
3568:
3566:
3565:
3560:
3539:
3537:
3536:
3531:
3502:
3500:
3499:
3494:
3483:
3482:
3464:, the pre-image
3455:
3453:
3452:
3447:
3423:
3421:
3419:
3418:
3413:
3411:
3410:
3398:
3397:
3379:
3377:
3376:
3371:
3359:
3357:
3355:
3354:
3349:
3347:
3346:
3331:
3330:
3306:
3304:
3302:
3301:
3296:
3282:
3280:
3279:
3274:
3269:
3268:
3256:
3255:
3214:
3212:
3211:
3206:
3175:
3173:
3171:
3170:
3165:
3126:
3124:
3122:
3121:
3116:
3090:
3088:
3087:
3082:
3055:
3053:
3052:
3047:
3029:
3027:
3026:
3021:
3009:
3007:
3006:
3001:
2996:
2978:
2976:
2974:
2973:
2968:
2939:
2937:
2936:
2931:
2915:
2913:
2912:
2907:
2905:
2893:
2891:
2890:
2885:
2883:
2871:
2869:
2868:
2863:
2848:
2846:
2845:
2840:
2835:
2810:
2808:
2807:
2802:
2790:
2788:
2787:
2782:
2764:
2762:
2760:
2759:
2754:
2728:
2726:
2724:
2723:
2718:
2704:
2702:
2701:
2696:
2678:
2676:
2675:
2670:
2659:
2658:
2642:
2640:
2639:
2634:
2622:
2620:
2619:
2614:
2603:with entries in
2599:
2597:
2596:
2591:
2573:
2571:
2569:
2568:
2563:
2537:
2535:
2533:
2532:
2527:
2513:
2511:
2510:
2505:
2490:
2488:
2486:
2485:
2480:
2469:
2455:
2453:
2452:
2447:
2439:
2438:
2411:
2409:
2408:
2403:
2401:
2385:
2383:
2382:
2377:
2375:
2363:
2361:
2359:
2358:
2353:
2351:
2334:
2332:
2331:
2326:
2314:
2312:
2310:
2309:
2304:
2302:
2285:
2283:
2282:
2277:
2275:
2256:
2254:
2253:
2248:
2194:
2192:
2191:
2186:
2174:
2172:
2171:
2166:
2145:
2143:
2141:
2140:
2135:
2121:
2119:
2118:
2113:
2092:
2090:
2089:
2084:
2057:
2055:
2054:
2049:
2034:
2032:
2031:
2026:
2014:
2012:
2011:
2006:
1972:
1970:
1969:
1964:
1926:
1924:
1922:
1921:
1916:
1896:
1894:
1893:
1888:
1886:
1885:
1870:
1869:
1838:
1836:
1835:
1830:
1828:
1827:
1808:
1806:
1805:
1800:
1798:
1797:
1773:
1771:
1769:
1768:
1763:
1733:
1731:
1730:
1725:
1720:
1719:
1701:. Also, the set
1700:
1698:
1696:
1695:
1690:
1688:
1687:
1669:
1667:
1666:
1661:
1604:
1602:
1601:
1596:
1578:
1576:
1575:
1570:
1540:
1538:
1537:
1532:
1514:
1510:
1506:
1502:
1500:
1499:
1494:
1476:
1474:
1473:
1468:
1450:
1446:
1444:
1443:
1438:
1423:
1416:
1386:
1379:
1377:
1376:
1371:
1366:
1342:
1340:
1339:
1334:
1329:
1314:
1299:
1295:
1285:
1283:
1282:
1277:
1272:
1253:
1242:
1240:
1239:
1234:
1207:
1205:
1204:
1199:
1181:
1159:
1157:
1155:
1154:
1149:
1126:
1124:
1123:
1118:
1086:
1074:
1072:
1070:
1069:
1064:
1050:
1048:
1047:
1042:
1027:
1025:
1023:
1022:
1017:
997:
995:
994:
989:
968:
966:
964:
963:
958:
928:
926:
925:
920:
893:
891:
890:
885:
873:
871:
869:
868:
863:
849:
847:
846:
841:
821:
817:
809:
775:Richard Dedekind
751:fractional ideal
644:
637:
630:
615:Operator algebra
601:Clifford algebra
537:
535:
534:
529:
524:
523:
511:
490:
488:
487:
482:
480:
479:
474:
453:
451:
450:
445:
443:
442:
437:
415:Ring of integers
409:
406:Integers modulo
357:Euclidean domain
245:
243:
242:
237:
235:
227:
222:
199:
197:
196:
191:
189:
93:Product of rings
79:Fractional ideal
38:
30:
29:
21:
14116:
14115:
14111:
14110:
14109:
14107:
14106:
14105:
14076:
14075:
14057:
14052:
14030:
14004:
13982:
13972:Springer-Verlag
13949:
13931:Springer-Verlag
13919:Eisenbud, David
13911:
13892:
13866:
13865:
13857:
13853:
13844:
13842:
13829:
13828:
13824:
13815:
13813:
13800:
13799:
13795:
13786:
13784:
13771:
13770:
13766:
13758:
13754:
13746:
13742:
13735:
13716:
13712:
13704:
13700:
13692:
13688:
13675:
13674:
13670:
13656:
13652:
13644:
13640:
13636:, Section III.2
13632:
13628:
13620:
13609:
13601:
13597:
13589:
13585:
13577:
13573:
13565:
13561:
13553:
13549:
13538:
13534:
13523:
13519:
13508:
13501:
13496:
13491:
13490:
13455:
13437:
13428:with things in
13419:
13415:
13398:
13394:
13389:
13342:
13319:
13317:
13314:
13313:
13311:
13293:
13285:
13282:
13281:
13279:
13263:
13258:
13255:
13254:
13236:
13234:
13231:
13230:
13228:
13212:
13210:
13207:
13206:
13188:
13180:
13177:
13176:
13174:
13153:
13150:
13149:
13147:
13131:
13128:
13127:
13111:
13108:
13107:
13089:
13086:
13085:
13083:
13067:
13064:
13063:
13047:
13044:
13043:
13027:
13024:
13023:
13005:
13002:
13001:
12999:
12979:
12976:
12975:
12959:
12956:
12955:
12939:
12936:
12935:
12927:
12922:is used alone.
12903:
12900:
12899:
12896:two-sided ideal
12853:
12850:
12849:
12847:
12805:
12802:
12801:
12799:
12761:
12758:
12757:
12755:
12733:
12730:
12729:
12693:
12690:
12689:
12687:
12653:
12650:
12649:
12631:
12628:
12627:
12607:
12604:
12603:
12580:
12577:
12576:
12558:
12555:
12554:
12552:
12536:
12533:
12532:
12513:
12510:
12509:
12481:
12478:
12477:
12447:
12444:
12443:
12441:
12435:
12433:Generalizations
12411:
12410:
12408:
12405:
12404:
12402:
12385:
12384:
12382:
12379:
12378:
12358:
12357:
12353:
12347:
12341:
12340:
12339:
12337:
12334:
12333:
12331:
12313:
12312:
12308:
12302:
12301:
12299:
12296:
12295:
12276:
12275:
12267:
12264:
12263:
12261:
12240:
12239:
12237:
12234:
12233:
12214:
12213:
12205:
12202:
12201:
12199:
12177:
12176:
12172:
12170:
12167:
12166:
12147:
12146:
12138:
12135:
12134:
12132:
12115:
12109:
12108:
12107:
12097:
12096:
12092:
12082:
12081:
12077:
12071:
12065:
12064:
12063:
12061:
12058:
12057:
12035:
12029:
12028:
12027:
12018:
12017:
12015:
12012:
12011:
12009:
11992:
11991:
11989:
11986:
11985:
11957:
11956:
11947:
11946:
11944:
11941:
11940:
11882:field extension
11849:
11843:
11842:
11841:
11839:
11836:
11835:
11820:
11817:
11816:
11792:
11791:
11789:
11786:
11785:
11762:
11756:
11755:
11754:
11752:
11749:
11748:
11733:
11730:
11729:
11705:
11704:
11702:
11699:
11698:
11677:
11676:
11664:
11658:
11657:
11656:
11654:
11651:
11650:
11648:
11631:
11630:
11618:
11612:
11611:
11610:
11608:
11605:
11604:
11571:
11570:
11568:
11565:
11564:
11533:
11529:
11508:
11504:
11496:
11493:
11492:
11490:
11468:
11464:
11411:
11408:
11407:
11405:
11383:
11379:
11326:
11323:
11322:
11320:
11291:
11287:
11273:
11270:
11269:
11248:
11244:
11236:
11233:
11232:
11194:
11191:
11190:
11141:
11138:
11137:
11115:
11110:
11102:
11099:
11098:
11074:
11069:
11061:
11059:
11056:
11055:
11053:
11029:
11023:
11022:
11021:
11019:
11016:
11015:
10994:
10993:
10991:
10988:
10987:
10967:
10966:
10954:
10948:
10947:
10946:
10944:
10941:
10940:
10922:
10921:
10909:
10903:
10902:
10901:
10899:
10896:
10895:
10872:
10866:
10865:
10864:
10862:
10859:
10858:
10843:
10840:
10839:
10819:
10818:
10816:
10813:
10812:
10804:is an ideal in
10788:
10787:
10785:
10782:
10781:
10776:is an ideal in
10760:
10759:
10757:
10754:
10753:
10717:
10716:
10714:
10711:
10710:
10693:
10687:
10686:
10685:
10683:
10680:
10679:
10652:
10651:
10639:
10635:
10633:
10630:
10629:
10624:is an ideal of
10608:
10607:
10605:
10602:
10601:
10581:
10580:
10568:
10564:
10555:
10554:
10545:
10541:
10529:
10525:
10513:
10509:
10500:
10499:
10490:
10484:
10483:
10482:
10480:
10477:
10476:
10451:
10450:
10442:
10439:
10438:
10436:
10411:
10410:
10408:
10405:
10404:
10387:
10381:
10380:
10379:
10377:
10374:
10373:
10330:
10329:
10321:
10318:
10317:
10312:is an ideal in
10296:
10295:
10293:
10290:
10289:
10258:
10228:
10227:
10215:
10211:
10202:
10201:
10195:
10191:
10189:
10186:
10185:
10183:
10154:
10150:
10136:
10130:
10129:
10118:
10115:
10114:
10094:
10090:
10072:
10071:
10069:
10066:
10065:
10038:
10034:
10025:
10021:
10019:
10016:
10015:
9967:
9964:
9963:
9961:
9933:
9930:
9929:
9882:
9879:
9878:
9832:
9831:
9824:
9818:
9817:
9816:
9792:
9789:
9788:
9757:
9754:
9753:
9751:
9722:
9719:
9718:
9671:
9668:
9667:
9665:
9637:
9623:
9620:
9619:
9595:
9592:
9591:
9589:
9554:
9551:
9550:
9548:
9512:
9509:
9508:
9507:if and only if
9479:
9476:
9475:
9441:
9405:
9402:
9401:
9379:
9378:
9373:
9368:
9365:
9364:
9362:
9341:
9340:
9338:
9335:
9334:
9317:
9316:
9314:
9311:
9310:
9282:
9279:
9278:
9241:
9215:
9210:
9207:
9206:
9181:
9178:
9177:
9153:
9150:
9149:
9126:
9125:
9118:
9112:
9111:
9110:
9098:
9095:
9094:
9053:
9050:
9049:
9031:primitive ideal
9013:
9007:
8977:
8976:
8970:
8969:
8957:
8953:
8926:
8925:
8916:
8915:
8913:
8910:
8909:
8892:
8891:
8885:
8884:
8875:
8874:
8865:
8864:
8862:
8859:
8858:
8837:
8833:
8797:
8793:
8772:
8771:
8765:
8764:
8762:
8759:
8758:
8737:
8733:
8670:
8666:
8564:
8563:
8557:
8556:
8554:
8551:
8550:
8508:
8507:
8498:
8497:
8495:
8492:
8491:
8402:
8401:
8392:
8391:
8389:
8386:
8385:
8339:
8338:
8299:
8298:
8271:
8270:
8268:
8265:
8264:
8262:
8219:
8211:
8208:
8207:
8186:
8183:
8182:
8180:
8164:
8161:
8160:
8126:
8123:
8122:
8103:
8056:
8053:
8052:
8033:
8031:
8028:
8027:
8024:
7996:
7995:
7989:
7988:
7978:
7977:
7976:
7974:
7971:
7970:
7968:
7951:
7950:
7948:
7945:
7944:
7927:
7926:
7917:
7916:
7914:
7911:
7910:
7907:Dedekind domain
7884:
7883:
7877:
7876:
7871:
7862:
7861:
7852:
7851:
7836:
7835:
7830:
7818:
7817:
7812:
7797:
7792:
7786:
7783:
7782:
7780:
7753:
7752:
7750:
7747:
7746:
7744:
7723:
7722:
7720:
7717:
7716:
7686:
7685:
7676:
7675:
7673:
7670:
7669:
7649:
7648:
7642:
7641:
7632:
7631:
7622:
7621:
7619:
7616:
7615:
7594:
7593:
7584:
7583:
7581:
7578:
7577:
7563:modular lattice
7536:
7535:
7533:
7530:
7529:
7512:
7511:
7509:
7506:
7505:
7488:
7487:
7485:
7482:
7481:
7461:
7460:
7451:
7450:
7441:
7440:
7431:
7430:
7418:
7417:
7408:
7407:
7395:
7394:
7392:
7389:
7388:
7362:
7361:
7355:
7354:
7345:
7344:
7338:
7337:
7328:
7327:
7318:
7317:
7308:
7307:
7302:
7299:
7298:
7296:
7275:
7274:
7268:
7267:
7258:
7257:
7251:
7250:
7238:
7237:
7228:
7227:
7218:
7217:
7215:
7212:
7211:
7209:
7187:
7186:
7177:
7176:
7167:
7166:
7164:
7161:
7160:
7158:
7136:
7135:
7133:
7130:
7129:
7127:
7110:
7109:
7107:
7104:
7103:
7086:
7085:
7079:
7078:
7076:
7073:
7072:
7053:
7052:
7043:
7042:
7040:
7037:
7036:
7034:
7017:
7016:
7014:
7011:
7010:
6993:
6992:
6990:
6987:
6986:
6969:
6968:
6959:
6958:
6956:
6953:
6952:
6930:
6929:
6927:
6924:
6923:
6921:
6900:
6899:
6897:
6894:
6893:
6840: for
6837:
6798:
6797:
6788:
6784:
6780: and
6777:
6771:
6770:
6761:
6757:
6748:
6744:
6738:
6734:
6719:
6715:
6709:
6705:
6693:
6692:
6686:
6685:
6683:
6680:
6679:
6675:are two-sided,
6659:
6658:
6649:
6648:
6646:
6643:
6642:
6618:
6617:
6607: and
6604:
6598:
6597:
6567:
6566:
6557:
6556:
6554:
6551:
6550:
6546:, their sum is
6524:
6523:
6521:
6518:
6517:
6515:
6498:
6497:
6495:
6492:
6491:
6488:
6468:
6446:
6383:equidimensional
6335:
6332:
6331:
6327:
6291:
6273:
6272:
6254:
6253:
6251:
6248:
6247:
6245:
6217:
6214:
6213:
6209:
6193:Parameter ideal
6185:Nilpotent ideal
6141:
6138:
6137:
6135:
6113:
6110:
6109:
6081:
6078:
6077:
6072:are said to be
6069:
6065:
6035:Primitive ideal
6013:Principal ideal
5998:
5986:
5962:
5942:
5930:
5906:
5903:semiprime ideal
5891:integral domain
5868:
5865:
5864:
5862:
5846:
5843:
5842:
5826:
5823:
5822:
5804:
5801:
5800:
5798:
5779:
5776:
5775:
5757:
5754:
5753:
5751:
5735:
5732:
5731:
5715:
5712:
5711:
5691:
5688:
5687:
5663:
5660:
5659:
5631:
5628:
5627:
5597:
5585:
5564:
5562:Types of ideals
5540:
5537:
5536:
5534:
5533:is an ideal of
5518:
5515:
5514:
5466:
5463:
5462:
5460:
5442:
5439:
5438:
5436:
5420:
5417:
5416:
5398:
5395:
5394:
5392:
5376:
5373:
5372:
5342:
5339:
5338:
5336:
5314:
5311:
5310:
5292:
5289:
5288:
5286:
5270:
5267:
5266:
5237:
5233:
5218:
5214:
5209:
5206:
5205:
5203:
5181:
5178:
5177:
5157:
5153:
5138:
5134:
5123:
5120:
5119:
5095:
5092:
5091:
5089:
5067:
5064:
5063:
5030:
5027:
5026:
5024:
5005:
5002:
5001:
4957:
4953:
4938:
4934:
4919:
4915:
4907:
4892:
4888:
4882:
4878:
4872:
4868:
4853:
4849:
4843:
4839:
4833:
4829:
4812:
4809:
4808:
4763:
4759:
4744:
4740:
4732:
4717:
4713:
4707:
4703:
4688:
4684:
4678:
4674:
4660:
4657:
4656:
4645:of elements of
4619:
4616:
4615:
4590:
4587:
4586:
4584:
4562:Krull's theorem
4531:
4528:
4527:
4525:
4496:
4490:
4489:
4488:
4486:
4483:
4482:
4460:
4457:
4456:
4431:
4425:
4424:
4423:
4421:
4418:
4417:
4390:
4384:
4383:
4382:
4380:
4377:
4376:
4354:
4351:
4350:
4317:
4311:
4310:
4309:
4297:
4290:
4287:
4286:
4262:
4259:
4258:
4256:
4239:
4233:
4232:
4231:
4222:
4216:
4215:
4214:
4212:
4209:
4208:
4192:
4189:
4188:
4179:ascending chain
4148:
4142:
4141:
4140:
4138:
4135:
4134:
4105:
4104:
4095:
4094:
4089:
4086:
4085:
4083:
4066:
4065:
4063:
4060:
4059:
4042:
4041:
4039:
4036:
4035:
4026:is an ideal of
4010:
4009:
4004:
3995:
3994:
3985:
3984:
3979:
3976:
3975:
3950:
3949:
3940:
3939:
3937:
3934:
3933:
3855:
3851:
3849:
3846:
3845:
3820:
3817:
3816:
3802:Types of ideals
3770:
3766:
3758:
3755:
3754:
3752:
3721:
3718:
3717:
3701:
3698:
3697:
3681:
3678:
3677:
3661:
3658:
3657:
3627:
3624:
3623:
3621:
3582:
3579:
3578:
3577:is surjective,
3545:
3542:
3541:
3516:
3513:
3512:
3475:
3471:
3469:
3466:
3465:
3429:
3426:
3425:
3406:
3402:
3393:
3389:
3387:
3384:
3383:
3381:
3365:
3362:
3361:
3342:
3338:
3326:
3322:
3314:
3311:
3310:
3308:
3290:
3287:
3286:
3284:
3264:
3260:
3248:
3244:
3224:
3221:
3220:
3188:
3185:
3184:
3141:
3138:
3137:
3135:
3098:
3095:
3094:
3092:
3061:
3058:
3057:
3035:
3032:
3031:
3015:
3012:
3011:
2992:
2984:
2981:
2980:
2947:
2944:
2943:
2941:
2925:
2922:
2921:
2901:
2899:
2896:
2895:
2879:
2877:
2874:
2873:
2857:
2854:
2853:
2831:
2823:
2820:
2819:
2796:
2793:
2792:
2791:matrices whose
2770:
2767:
2766:
2736:
2733:
2732:
2730:
2712:
2709:
2708:
2706:
2684:
2681:
2680:
2654:
2650:
2648:
2645:
2644:
2628:
2625:
2624:
2608:
2605:
2604:
2579:
2576:
2575:
2545:
2542:
2541:
2539:
2521:
2518:
2517:
2515:
2499:
2496:
2495:
2465:
2463:
2460:
2459:
2457:
2434:
2430:
2428:
2425:
2424:
2419:The set of all
2397:
2395:
2392:
2391:
2371:
2369:
2366:
2365:
2347:
2342:
2339:
2338:
2336:
2320:
2317:
2316:
2298:
2293:
2290:
2289:
2287:
2271:
2269:
2266:
2265:
2200:
2197:
2196:
2180:
2177:
2176:
2151:
2148:
2147:
2129:
2126:
2125:
2123:
2098:
2095:
2094:
2063:
2060:
2059:
2040:
2037:
2036:
2020:
2017:
2016:
1982:
1979:
1978:
1940:
1937:
1936:
1904:
1901:
1900:
1898:
1881:
1880:
1862:
1858:
1844:
1841:
1840:
1823:
1822:
1814:
1811:
1810:
1793:
1792:
1790:
1787:
1786:
1751:
1748:
1747:
1745:
1739:
1715:
1711:
1706:
1703:
1702:
1683:
1679:
1677:
1674:
1673:
1671:
1649:
1646:
1645:
1622:
1584:
1581:
1580:
1546:
1543:
1542:
1520:
1517:
1516:
1512:
1508:
1504:
1482:
1479:
1478:
1456:
1453:
1452:
1448:
1429:
1426:
1425:
1421:
1414:
1407:
1384:
1362:
1351:
1348:
1347:
1325:
1320:
1317:
1316:
1312:
1297:
1293:
1286:and called the
1268:
1263:
1260:
1259:
1251:
1216:
1213:
1212:
1208:if and only if
1187:
1184:
1183:
1179:
1167:If the ring is
1162:two-sided ideal
1134:
1131:
1130:
1128:
1103:
1100:
1099:
1084:
1058:
1055:
1054:
1052:
1033:
1030:
1029:
1005:
1002:
1001:
999:
977:
974:
973:
940:
937:
936:
934:
902:
899:
898:
879:
876:
875:
857:
854:
853:
851:
835:
832:
831:
819:
815:
807:
801:
791:and especially
763:
755:integral ideals
732:Dedekind domain
711:principal ideal
696:normal subgroup
648:
619:
618:
551:
541:
540:
519:
515:
507:
505:
502:
501:
475:
470:
469:
467:
464:
463:
438:
433:
432:
430:
427:
426:
407:
377:Polynomial ring
327:Integral domain
316:
306:
305:
231:
223:
218:
210:
207:
206:
185:
183:
180:
179:
165:Involutive ring
50:
39:
33:
28:
23:
22:
15:
12:
11:
5:
14114:
14104:
14103:
14098:
14093:
14088:
14074:
14073:
14068:Stack Exchange
14056:
14055:External links
14053:
14051:
14050:
14028:
14008:
14002:
13986:
13980:
13960:
13947:
13915:
13909:
13896:
13890:
13867:
13864:
13863:
13851:
13822:
13793:
13764:
13752:
13740:
13733:
13710:
13708:, p. 251.
13698:
13696:, p. 255.
13686:
13683:. 22 Aug 2024.
13668:
13650:
13648:, p. 244.
13638:
13626:
13624:, p. 243.
13607:
13595:
13583:
13571:
13559:
13547:
13532:
13517:
13514:. p. 439.
13498:
13497:
13495:
13492:
13489:
13488:
13413:
13405:trivial ideals
13391:
13390:
13388:
13385:
13384:
13383:
13378:
13373:
13368:
13363:
13358:
13353:
13348:
13341:
13338:
13322:
13296:
13292:
13289:
13266:
13262:
13239:
13215:
13205:, an ideal of
13191:
13187:
13184:
13157:
13135:
13115:
13093:
13071:
13051:
13031:
13009:
12983:
12963:
12943:
12907:
12881:
12878:
12875:
12872:
12869:
12866:
12863:
12860:
12857:
12833:
12830:
12827:
12824:
12821:
12818:
12815:
12812:
12809:
12792:
12791:
12777:
12774:
12771:
12768:
12765:
12743:
12740:
12737:
12728:, the product
12715:
12712:
12709:
12706:
12703:
12700:
12697:
12675:
12672:
12669:
12666:
12663:
12660:
12657:
12646:
12635:
12611:
12584:
12562:
12540:
12517:
12485:
12463:
12460:
12457:
12454:
12451:
12434:
12431:
12414:
12388:
12361:
12356:
12350:
12344:
12316:
12311:
12305:
12279:
12274:
12271:
12243:
12217:
12212:
12209:
12180:
12175:
12150:
12145:
12142:
12118:
12112:
12106:
12100:
12095:
12091:
12085:
12080:
12074:
12068:
12041:
12038:
12032:
12026:
12021:
11995:
11960:
11955:
11950:
11871:
11870:
11852:
11846:
11824:
11795:
11783:
11765:
11759:
11737:
11708:
11696:
11680:
11675:
11670:
11667:
11661:
11634:
11629:
11624:
11621:
11615:
11588:
11585:
11582:
11579:
11574:
11536:
11532:
11528:
11525:
11522:
11519:
11516:
11511:
11507:
11503:
11500:
11476:
11471:
11467:
11463:
11460:
11457:
11454:
11451:
11448:
11445:
11442:
11439:
11436:
11433:
11430:
11427:
11424:
11421:
11418:
11415:
11391:
11386:
11382:
11378:
11375:
11372:
11369:
11366:
11363:
11360:
11357:
11354:
11351:
11348:
11345:
11342:
11339:
11336:
11333:
11330:
11308:
11305:
11302:
11299:
11294:
11290:
11286:
11283:
11280:
11277:
11251:
11247:
11243:
11240:
11216:
11213:
11210:
11207:
11204:
11201:
11198:
11178:
11175:
11172:
11169:
11166:
11163:
11160:
11157:
11154:
11151:
11148:
11145:
11124:
11121:
11118:
11113:
11109:
11106:
11083:
11080:
11077:
11072:
11068:
11064:
11032:
11026:
10997:
10984:
10983:
10970:
10965:
10960:
10957:
10951:
10938:
10925:
10920:
10915:
10912:
10906:
10893:
10875:
10869:
10847:
10822:
10791:
10763:
10720:
10696:
10690:
10660:
10655:
10650:
10645:
10642:
10638:
10611:
10598:
10597:
10584:
10579:
10576:
10571:
10567:
10563:
10558:
10553:
10548:
10544:
10540:
10537:
10532:
10528:
10524:
10521:
10516:
10512:
10508:
10503:
10498:
10493:
10487:
10472:. Explicitly,
10459:
10454:
10449:
10446:
10414:
10390:
10384:
10338:
10333:
10328:
10325:
10299:
10257:
10254:
10239:
10236:
10231:
10224:
10221:
10218:
10214:
10210:
10205:
10198:
10194:
10171:
10168:
10165:
10162:
10157:
10153:
10149:
10146:
10143:
10139:
10133:
10128:
10125:
10122:
10102:
10097:
10093:
10089:
10086:
10083:
10080:
10075:
10047:
10044:
10041:
10037:
10033:
10028:
10024:
10001:
9998:
9995:
9992:
9989:
9986:
9983:
9980:
9977:
9974:
9971:
9949:
9946:
9943:
9940:
9937:
9898:
9895:
9892:
9889:
9886:
9867:
9866:
9855:
9852:
9849:
9846:
9843:
9840:
9835:
9821:
9815:
9811:
9808:
9805:
9802:
9799:
9796:
9782:
9781:
9767:
9764:
9761:
9735:
9732:
9729:
9726:
9696:
9693:
9690:
9687:
9684:
9681:
9678:
9675:
9653:
9650:
9647:
9644:
9640:
9636:
9633:
9630:
9627:
9605:
9602:
9599:
9567:
9564:
9561:
9558:
9525:
9522:
9519:
9516:
9492:
9489:
9486:
9483:
9472:
9471:
9460:
9457:
9454:
9451:
9448:
9439:
9436:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9412:
9409:
9382:
9376:
9372:
9344:
9320:
9298:
9295:
9292:
9289:
9286:
9266:
9263:
9260:
9257:
9254:
9251:
9248:
9244:
9240:
9237:
9234:
9231:
9228:
9225:
9222:
9218:
9214:
9194:
9191:
9188:
9185:
9157:
9146:
9145:
9134:
9129:
9115:
9109:
9105:
9102:
9075:
9072:
9069:
9066:
9063:
9060:
9057:
9023:
9022:
9009:Main article:
9006:
9003:
8994:
8993:
8980:
8973:
8968:
8965:
8960:
8956:
8952:
8949:
8946:
8943:
8940:
8937:
8934:
8929:
8924:
8919:
8895:
8888:
8883:
8878:
8873:
8868:
8856:
8845:
8840:
8836:
8832:
8829:
8826:
8823:
8820:
8817:
8814:
8811:
8808:
8805:
8800:
8796:
8792:
8789:
8786:
8783:
8780:
8775:
8768:
8756:
8745:
8740:
8736:
8732:
8729:
8726:
8723:
8720:
8717:
8714:
8711:
8708:
8705:
8702:
8699:
8696:
8693:
8690:
8687:
8684:
8681:
8678:
8673:
8669:
8665:
8662:
8659:
8656:
8653:
8650:
8647:
8644:
8641:
8638:
8635:
8632:
8629:
8626:
8623:
8620:
8617:
8614:
8611:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8584:
8581:
8578:
8575:
8572:
8567:
8560:
8548:
8537:
8534:
8531:
8528:
8525:
8522:
8519:
8516:
8511:
8506:
8501:
8479:
8476:
8473:
8470:
8467:
8464:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8434:
8431:
8428:
8425:
8422:
8419:
8416:
8413:
8410:
8405:
8400:
8395:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8342:
8337:
8334:
8331:
8328:
8325:
8322:
8319:
8316:
8313:
8310:
8307:
8302:
8297:
8294:
8291:
8288:
8285:
8282:
8279:
8274:
8250:
8247:
8244:
8241:
8238:
8235:
8232:
8229:
8226:
8222:
8218:
8215:
8190:
8168:
8148:
8145:
8142:
8139:
8136:
8133:
8130:
8119:
8118:
8106:
8102:
8099:
8096:
8093:
8090:
8087:
8084:
8081:
8078:
8075:
8072:
8069:
8066:
8063:
8060:
8036:
8023:
8020:
7999:
7992:
7987:
7981:
7954:
7930:
7925:
7920:
7887:
7880:
7874:
7870:
7865:
7860:
7855:
7850:
7847:
7844:
7839:
7833:
7829:
7826:
7821:
7815:
7811:
7808:
7805:
7800:
7795:
7791:
7773:
7772:
7756:
7726:
7714:
7703:
7700:
7697:
7694:
7689:
7684:
7679:
7652:
7645:
7640:
7635:
7630:
7625:
7597:
7592:
7587:
7539:
7515:
7491:
7478:
7477:
7464:
7459:
7454:
7449:
7444:
7439:
7434:
7429:
7426:
7421:
7416:
7411:
7406:
7403:
7398:
7382:
7381:
7365:
7358:
7353:
7348:
7341:
7336:
7331:
7326:
7321:
7316:
7311:
7306:
7294:
7278:
7271:
7266:
7261:
7254:
7249:
7246:
7241:
7236:
7231:
7226:
7221:
7190:
7185:
7180:
7175:
7170:
7139:
7113:
7089:
7082:
7056:
7051:
7046:
7033:(or the union
7020:
6996:
6972:
6967:
6962:
6933:
6903:
6882:
6881:
6870:
6867:
6864:
6861:
6858:
6855:
6852:
6849:
6846:
6836:
6833:
6830:
6827:
6824:
6821:
6818:
6815:
6812:
6809:
6806:
6801:
6796:
6791:
6787:
6774:
6769:
6764:
6760:
6756:
6751:
6747:
6741:
6737:
6733:
6730:
6727:
6722:
6718:
6712:
6708:
6704:
6701:
6696:
6689:
6662:
6657:
6652:
6639:
6638:
6626:
6621:
6616:
6613:
6601:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6570:
6565:
6560:
6527:
6501:
6487:
6484:
6483:
6482:
6430:
6417:submodules of
6408:quotient field
6391:
6390:
6339:
6319:
6301:
6298:
6294:
6290:
6287:
6284:
6281:
6271:
6268:
6265:
6262:
6221:
6201:
6189:
6181:
6173:
6165:
6151:
6148:
6145:
6123:
6120:
6117:
6097:
6094:
6091:
6088:
6085:
6059:
6051:
6031:
6021:
6009:
6003:natural number
5954:
5947:semiprime ring
5894:
5872:
5850:
5830:
5808:
5786:
5783:
5761:
5739:
5719:
5695:
5679:
5667:
5653:
5641:
5638:
5635:
5621:
5613:
5563:
5560:
5559:
5558:
5544:
5522:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5470:
5446:
5424:
5402:
5380:
5358:
5355:
5352:
5349:
5346:
5324:
5321:
5318:
5296:
5274:
5259:
5245:
5240:
5236:
5232:
5229:
5226:
5221:
5217:
5213:
5191:
5188:
5185:
5165:
5160:
5156:
5152:
5149:
5146:
5141:
5137:
5133:
5130:
5127:
5105:
5102:
5099:
5077:
5074:
5071:
5049:
5046:
5043:
5040:
5037:
5034:
5012:
5009:
4989:
4988:
4987:
4986:
4974:
4971:
4968:
4965:
4960:
4956:
4952:
4949:
4946:
4941:
4937:
4933:
4930:
4927:
4922:
4918:
4914:
4910:
4906:
4903:
4900:
4895:
4891:
4885:
4881:
4875:
4871:
4867:
4864:
4861:
4856:
4852:
4846:
4842:
4836:
4832:
4828:
4825:
4822:
4819:
4816:
4794:
4793:
4792:
4791:
4780:
4777:
4774:
4771:
4766:
4762:
4758:
4755:
4752:
4747:
4743:
4739:
4735:
4731:
4728:
4725:
4720:
4716:
4710:
4706:
4702:
4699:
4696:
4691:
4687:
4681:
4677:
4673:
4670:
4667:
4664:
4639:(finite) left
4626:
4623:
4597:
4594:
4565:
4547:
4544:
4541:
4538:
4535:
4513:
4510:
4507:
4504:
4499:
4493:
4470:
4467:
4464:
4434:
4428:
4401:
4398:
4393:
4387:
4364:
4361:
4358:
4343:
4320:
4314:
4306:
4303:
4300:
4296:
4272:
4269:
4266:
4242:
4236:
4230:
4225:
4219:
4196:
4181:of left ideals
4165:
4162:
4159:
4156:
4151:
4145:
4131:
4113:
4108:
4103:
4098:
4093:
4069:
4045:
4032:ideal quotient
4013:
4007:
4003:
3998:
3993:
3988:
3983:
3953:
3948:
3943:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3858:
3854:
3830:
3827:
3824:
3805:
3787:
3784:
3781:
3776:
3773:
3769:
3765:
3762:
3740:
3737:
3734:
3731:
3728:
3725:
3705:
3685:
3665:
3643:
3640:
3637:
3634:
3631:
3615:
3595:
3592:
3589:
3586:
3558:
3555:
3552:
3549:
3529:
3526:
3523:
3520:
3492:
3489:
3486:
3481:
3478:
3474:
3445:
3442:
3439:
3436:
3433:
3409:
3405:
3401:
3396:
3392:
3369:
3360:, and thus if
3345:
3341:
3337:
3334:
3329:
3325:
3321:
3318:
3294:
3272:
3267:
3263:
3259:
3254:
3251:
3247:
3243:
3240:
3237:
3234:
3231:
3228:
3204:
3201:
3198:
3195:
3192:
3181:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3128:
3114:
3111:
3108:
3105:
3102:
3080:
3077:
3074:
3071:
3068:
3065:
3045:
3042:
3039:
3019:
2999:
2995:
2991:
2988:
2966:
2963:
2960:
2957:
2954:
2951:
2929:
2904:
2882:
2861:
2838:
2834:
2830:
2827:
2816:
2800:
2780:
2777:
2774:
2752:
2749:
2746:
2743:
2740:
2716:
2694:
2691:
2688:
2668:
2665:
2662:
2657:
2653:
2632:
2612:
2589:
2586:
2583:
2561:
2558:
2555:
2552:
2549:
2525:
2503:
2492:
2478:
2475:
2472:
2468:
2445:
2442:
2437:
2433:
2417:
2400:
2374:
2350:
2346:
2324:
2301:
2297:
2274:
2258:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2184:
2164:
2161:
2158:
2155:
2133:
2111:
2108:
2105:
2102:
2082:
2079:
2076:
2073:
2070:
2067:
2047:
2044:
2024:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1914:
1911:
1908:
1884:
1879:
1876:
1873:
1868:
1865:
1861:
1857:
1854:
1851:
1848:
1826:
1821:
1818:
1796:
1775:
1761:
1758:
1755:
1735:
1723:
1718:
1714:
1710:
1686:
1682:
1659:
1656:
1653:
1621:
1618:
1594:
1591:
1588:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1530:
1527:
1524:
1492:
1489:
1486:
1466:
1463:
1460:
1436:
1433:
1406:
1403:
1381:
1380:
1369:
1365:
1361:
1358:
1355:
1332:
1328:
1324:
1315:is two-sided,
1275:
1271:
1267:
1244:
1243:
1232:
1229:
1226:
1223:
1220:
1197:
1194:
1191:
1147:
1144:
1141:
1138:
1116:
1113:
1110:
1107:
1077:
1076:
1062:
1040:
1037:
1028:, the product
1015:
1012:
1009:
987:
984:
981:
970:
956:
953:
950:
947:
944:
918:
915:
912:
909:
906:
883:
861:
839:
828:additive group
800:
797:
762:
759:
700:quotient group
678:, such as the
650:
649:
647:
646:
639:
632:
624:
621:
620:
612:
611:
583:
582:
576:
570:
564:
552:
547:
546:
543:
542:
539:
538:
527:
522:
518:
514:
510:
491:
478:
473:
454:
441:
436:
424:-adic integers
417:
411:
402:
388:
387:
386:
385:
379:
373:
372:
371:
359:
353:
347:
341:
335:
317:
312:
311:
308:
307:
304:
303:
302:
301:
289:
288:
287:
281:
269:
268:
267:
249:
248:
247:
246:
234:
230:
226:
221:
217:
214:
200:
188:
167:
161:
155:
149:
135:
134:
128:
122:
108:
107:
101:
95:
89:
88:
87:
81:
69:
63:
51:
49:Basic concepts
48:
47:
44:
43:
26:
9:
6:
4:
3:
2:
14113:
14102:
14099:
14097:
14094:
14092:
14089:
14087:
14084:
14083:
14081:
14070:
14069:
14064:
14059:
14058:
14047:
14043:
14039:
14035:
14031:
14029:9780691081014
14025:
14021:
14017:
14013:
14009:
14005:
14003:1-4020-2690-0
13999:
13995:
13991:
13987:
13983:
13977:
13973:
13969:
13965:
13961:
13958:
13954:
13950:
13944:
13940:
13936:
13932:
13928:
13924:
13920:
13916:
13912:
13910:9780471433347
13906:
13902:
13897:
13893:
13891:0-201-00361-9
13887:
13883:
13882:
13877:
13873:
13869:
13868:
13860:
13855:
13841:on 2017-01-16
13840:
13836:
13832:
13826:
13812:on 2017-01-16
13811:
13807:
13803:
13797:
13783:on 2017-01-16
13782:
13778:
13774:
13768:
13761:
13760:Milnor (1971)
13756:
13749:
13748:Eisenbud 1995
13744:
13736:
13734:9781139171762
13730:
13726:
13725:
13720:
13714:
13707:
13702:
13695:
13690:
13682:
13678:
13672:
13665:. p. 39.
13664:
13663:
13654:
13647:
13642:
13635:
13630:
13623:
13618:
13616:
13614:
13612:
13604:
13599:
13592:
13587:
13580:
13575:
13568:
13563:
13557:, p. 242
13556:
13551:
13544:. p. 83.
13543:
13536:
13529:. p. 76.
13528:
13521:
13513:
13506:
13504:
13499:
13485:
13481:
13477:
13473:
13467:
13463:
13459:
13453:
13448:
13444:
13440:
13435:
13431:
13427:
13423:
13417:
13410:
13406:
13402:
13396:
13392:
13382:
13379:
13377:
13374:
13372:
13369:
13367:
13364:
13362:
13359:
13357:
13354:
13352:
13349:
13347:
13344:
13343:
13337:
13290:
13287:
13260:
13185:
13182:
13171:
13155:
13146:-bimodule of
13133:
13113:
13091:
13069:
13049:
13029:
13007:
12997:
12981:
12961:
12941:
12933:
12930:
12923:
12921:
12905:
12897:
12876:
12873:
12870:
12864:
12861:
12858:
12855:
12828:
12825:
12822:
12816:
12813:
12810:
12807:
12797:
12772:
12769:
12766:
12741:
12738:
12735:
12710:
12707:
12704:
12698:
12695:
12670:
12667:
12664:
12658:
12655:
12647:
12633:
12625:
12609:
12602:
12601:
12600:
12598:
12582:
12560:
12538:
12531:
12515:
12507:
12503:
12499:
12483:
12458:
12455:
12452:
12440:
12439:monoid object
12430:
12354:
12348:
12309:
12272:
12269:
12259:
12210:
12207:
12197:
12173:
12143:
12140:
12116:
12093:
12089:
12078:
12072:
12039:
12036:
12024:
11982:
11980:
11976:
11953:
11939:
11935:
11931:
11927:
11926:inclusion map
11923:
11920:, and we let
11919:
11915:
11911:
11907:
11903:
11899:
11895:
11891:
11887:
11883:
11879:
11875:
11868:
11850:
11815:
11811:
11810:maximal ideal
11784:
11781:
11763:
11728:
11724:
11697:
11673:
11668:
11665:
11627:
11622:
11619:
11603:
11602:
11601:
11599:
11586:
11583:
11580:
11577:
11561:
11557:
11552:
11534:
11526:
11523:
11520:
11514:
11509:
11501:
11469:
11461:
11458:
11455:
11449:
11443:
11440:
11437:
11428:
11422:
11419:
11416:
11384:
11376:
11373:
11370:
11364:
11358:
11355:
11352:
11343:
11337:
11334:
11331:
11306:
11303:
11300:
11297:
11292:
11284:
11281:
11278:
11267:
11249:
11241:
11230:
11227:are units in
11214:
11211:
11208:
11205:
11202:
11199:
11196:
11173:
11170:
11167:
11158:
11155:
11152:
11146:
11143:
11122:
11119:
11116:
11107:
11104:
11081:
11078:
11075:
11052:
11048:
11030:
11014:implies that
11013:
10963:
10958:
10955:
10939:
10918:
10913:
10910:
10894:
10891:
10873:
10838:
10811:
10810:
10809:
10807:
10779:
10751:
10747:
10743:
10738:
10736:
10694:
10678:
10675:, called the
10674:
10643:
10640:
10636:
10627:
10577:
10574:
10569:
10565:
10561:
10551:
10546:
10542:
10538:
10530:
10526:
10519:
10514:
10510:
10506:
10496:
10491:
10475:
10474:
10473:
10444:
10435:generated by
10434:
10430:
10388:
10372:
10368:
10364:
10360:
10356:
10352:
10323:
10315:
10287:
10283:
10279:
10275:
10271:
10267:
10263:
10253:
10237:
10234:
10222:
10219:
10216:
10212:
10208:
10196:
10192:
10169:
10166:
10155:
10151:
10144:
10141:
10137:
10123:
10120:
10095:
10091:
10084:
10081:
10078:
10063:
10045:
10042:
10039:
10035:
10031:
10026:
10022:
9996:
9990:
9987:
9984:
9978:
9972:
9969:
9944:
9938:
9935:
9927:
9926:Artinian ring
9923:
9918:
9916:
9912:
9893:
9887:
9884:
9876:
9872:
9850:
9844:
9841:
9838:
9813:
9809:
9803:
9797:
9794:
9787:
9786:
9785:
9765:
9762:
9759:
9749:
9733:
9730:
9727:
9724:
9716:
9715:
9714:
9712:
9694:
9691:
9688:
9685:
9682:
9679:
9676:
9673:
9651:
9648:
9642:
9638:
9634:
9628:
9625:
9603:
9600:
9597:
9587:
9583:
9565:
9562:
9559:
9556:
9546:
9542:
9537:
9523:
9520:
9517:
9514:
9506:
9490:
9487:
9484:
9481:
9458:
9452:
9449:
9446:
9437:
9434:
9431:
9428:
9425:
9422:
9419:
9416:
9410:
9407:
9400:
9399:
9398:
9374:
9370:
9360:
9293:
9287:
9284:
9264:
9261:
9255:
9249:
9246:
9242:
9238:
9235:
9229:
9223:
9220:
9216:
9212:
9192:
9189:
9186:
9183:
9175:
9171:
9155:
9132:
9107:
9103:
9100:
9093:
9092:
9091:
9089:
9070:
9064:
9061:
9058:
9055:
9048:
9044:
9042:
9036:
9032:
9028:
9021:
9018:
9017:
9016:
9012:
9002:
9000:
8966:
8958:
8954:
8950:
8947:
8944:
8941:
8938:
8932:
8922:
8881:
8871:
8857:
8838:
8834:
8830:
8827:
8824:
8821:
8818:
8815:
8812:
8809:
8806:
8803:
8798:
8794:
8790:
8787:
8784:
8778:
8757:
8738:
8734:
8730:
8727:
8724:
8721:
8718:
8715:
8712:
8709:
8706:
8703:
8700:
8697:
8694:
8691:
8688:
8685:
8682:
8679:
8676:
8671:
8667:
8660:
8651:
8648:
8645:
8639:
8636:
8630:
8627:
8624:
8618:
8615:
8609:
8606:
8603:
8597:
8594:
8588:
8585:
8582:
8576:
8570:
8549:
8532:
8529:
8526:
8523:
8520:
8514:
8504:
8474:
8471:
8468:
8465:
8462:
8459:
8456:
8450:
8444:
8441:
8438:
8435:
8432:
8429:
8426:
8423:
8420:
8417:
8414:
8408:
8398:
8384:
8383:
8382:
8363:
8360:
8357:
8354:
8351:
8345:
8335:
8329:
8326:
8323:
8320:
8317:
8314:
8311:
8305:
8295:
8289:
8286:
8283:
8277:
8245:
8242:
8239:
8236:
8233:
8230:
8227:
8216:
8213:
8204:
8188:
8166:
8143:
8137:
8131:
8097:
8094:
8091:
8085:
8082:
8079:
8073:
8067:
8061:
8051:
8050:
8049:
8019:
8017:
7985:
7923:
7908:
7903:
7872:
7858:
7845:
7831:
7827:
7824:
7813:
7809:
7803:
7798:
7793:
7789:
7778:
7742:
7715:
7698:
7692:
7682:
7668:
7667:
7666:
7638:
7628:
7613:
7590:
7574:
7572:
7568:
7564:
7561:
7557:
7553:
7457:
7447:
7437:
7427:
7414:
7401:
7387:
7386:
7385:
7351:
7334:
7314:
7295:
7264:
7247:
7234:
7208:
7207:
7206:
7183:
7173:
7155:
7049:
6965:
6949:
6919:
6891:
6887:
6868:
6862:
6859:
6856:
6853:
6850:
6847:
6844:
6834:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6807:
6804:
6794:
6789:
6785:
6767:
6762:
6758:
6754:
6749:
6745:
6739:
6735:
6731:
6728:
6725:
6720:
6716:
6710:
6706:
6699:
6678:
6677:
6676:
6655:
6614:
6611:
6594:
6591:
6588:
6585:
6582:
6579:
6573:
6563:
6549:
6548:
6547:
6545:
6479:
6475:
6471:
6466:
6462:
6457:
6453:
6449:
6444:
6440:
6436:
6435:
6431:
6428:
6424:
6420:
6416:
6412:
6409:
6405:
6401:
6400:
6396:
6395:
6394:
6388:
6384:
6380:
6376:
6372:
6368:
6364:
6361:
6357:
6353:
6352:unmixed ideal
6350:is called an
6337:
6325:
6324:
6323:Unmixed ideal
6320:
6317:
6296:
6292:
6288:
6282:
6279:
6269:
6263:
6243:
6239:
6235:
6234:perfect ideal
6219:
6207:
6206:
6205:Perfect ideal
6202:
6199:
6195:
6194:
6190:
6187:
6186:
6182:
6179:
6178:
6174:
6171:
6170:
6169:Regular ideal
6166:
6149:
6146:
6143:
6121:
6118:
6115:
6095:
6092:
6089:
6086:
6083:
6075:
6064:: Two ideals
6063:
6060:
6057:
6056:
6052:
6049:
6045:
6041:
6037:
6036:
6032:
6029:
6025:
6022:
6019:
6015:
6014:
6010:
6007:
6004:
5996:
5992:
5984:
5980:
5976:
5972:
5968:
5967:primary ideal
5960:
5959:
5958:Primary ideal
5955:
5952:
5948:
5940:
5936:
5928:
5924:
5920:
5916:
5912:
5904:
5900:
5899:
5898:Radical ideal
5895:
5892:
5888:
5870:
5848:
5828:
5806:
5784:
5781:
5759:
5737:
5717:
5709:
5693:
5685:
5684:
5680:
5665:
5657:
5654:
5636:
5625:
5622:
5619:
5618:
5617:Minimal ideal
5614:
5611:
5607:
5603:
5595:
5591:
5590:maximal ideal
5583:
5582:
5581:Maximal ideal
5578:
5577:
5576:
5574:
5569:
5568:
5542:
5520:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5471:
5468:
5444:
5422:
5400:
5378:
5356:
5353:
5350:
5347:
5344:
5322:
5319:
5316:
5294:
5272:
5264:
5260:
5238:
5234:
5230:
5227:
5224:
5219:
5215:
5189:
5186:
5183:
5158:
5154:
5150:
5147:
5144:
5139:
5135:
5128:
5125:
5100:
5075:
5072:
5069:
5047:
5044:
5041:
5038:
5035:
5032:
5010:
5007:
4999:
4995:
4991:
4990:
4972:
4966:
4963:
4958:
4954:
4950:
4947:
4944:
4939:
4935:
4931:
4928:
4925:
4920:
4916:
4912:
4904:
4901:
4898:
4893:
4889:
4883:
4879:
4873:
4869:
4865:
4862:
4859:
4854:
4850:
4844:
4840:
4834:
4830:
4823:
4820:
4817:
4814:
4807:
4806:
4804:
4800:
4796:
4795:
4778:
4772:
4769:
4764:
4760:
4756:
4753:
4750:
4745:
4741:
4737:
4729:
4726:
4723:
4718:
4714:
4708:
4704:
4700:
4697:
4694:
4689:
4685:
4679:
4675:
4668:
4665:
4662:
4655:
4654:
4652:
4648:
4644:
4642:
4624:
4621:
4613:
4595:
4592:
4582:
4578:
4574:
4570:
4566:
4563:
4542:
4536:
4533:
4508:
4502:
4497:
4468:
4465:
4462:
4454:
4450:
4432:
4415:
4399:
4396:
4391:
4362:
4359:
4356:
4348:
4344:
4341:
4337:
4318:
4304:
4301:
4298:
4294:
4270:
4267:
4264:
4240:
4228:
4223:
4194:
4186:
4182:
4180:
4163:
4160:
4157:
4154:
4149:
4132:
4129:
4101:
4033:
4029:
4005:
3991:
3973:
3969:
3946:
3931:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3873:
3867:
3861:
3856:
3852:
3844:
3828:
3825:
3822:
3814:
3810:
3806:
3803:
3782:
3774:
3771:
3767:
3760:
3735:
3729:
3723:
3703:
3683:
3663:
3641:
3635:
3632:
3629:
3619:
3616:
3613:
3609:
3590:
3584:
3576:
3572:
3553:
3547:
3524:
3518:
3510:
3506:
3487:
3479:
3476:
3472:
3463:
3459:
3440:
3434:
3431:
3407:
3403:
3399:
3394:
3390:
3367:
3343:
3339:
3335:
3327:
3323:
3316:
3292:
3265:
3261:
3252:
3249:
3245:
3241:
3235:
3229:
3226:
3218:
3202:
3196:
3193:
3190:
3182:
3179:
3158:
3152:
3146:
3133:
3129:
3112:
3109:
3103:
3078:
3075:
3069:
3063:
3043:
3040:
3037:
3017:
2986:
2964:
2961:
2955:
2949:
2927:
2919:
2859:
2852:
2825:
2817:
2814:
2798:
2778:
2775:
2772:
2750:
2747:
2744:
2741:
2738:
2714:
2692:
2689:
2686:
2663:
2655:
2651:
2630:
2610:
2602:
2587:
2584:
2581:
2559:
2556:
2553:
2550:
2547:
2523:
2501:
2493:
2473:
2443:
2440:
2435:
2431:
2422:
2418:
2415:
2389:
2344:
2322:
2295:
2263:
2259:
2244:
2241:
2238:
2232:
2229:
2223:
2217:
2214:
2208:
2205:
2202:
2182:
2162:
2159:
2156:
2153:
2131:
2109:
2106:
2103:
2100:
2077:
2071:
2068:
2065:
2045:
2042:
2022:
1999:
1993:
1987:
1976:
1957:
1951:
1945:
1934:
1930:
1912:
1909:
1906:
1877:
1874:
1866:
1863:
1859:
1855:
1849:
1846:
1819:
1816:
1784:
1783:proper subset
1780:
1776:
1756:
1743:
1738:
1716:
1712:
1684:
1680:
1654:
1643:
1639:
1635:
1631:
1627:
1626:
1625:
1617:
1615:
1611:
1606:
1592:
1589:
1586:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1528:
1525:
1522:
1490:
1487:
1484:
1464:
1461:
1458:
1434:
1431:
1420:
1412:
1402:
1401:
1397:
1393:
1390:
1367:
1363:
1359:
1353:
1346:
1345:
1344:
1330:
1326:
1322:
1311:If the ideal
1309:
1307:
1303:
1291:
1290:
1273:
1269:
1265:
1257:
1249:
1230:
1227:
1224:
1221:
1218:
1211:
1210:
1209:
1195:
1192:
1189:
1176:
1174:
1170:
1165:
1163:
1145:
1142:
1139:
1136:
1114:
1111:
1108:
1105:
1097:
1092:
1091:over itself.
1090:
1082:
1060:
1038:
1035:
1013:
1010:
1007:
985:
982:
979:
971:
951:
948:
945:
932:
913:
910:
907:
897:
896:
895:
881:
859:
837:
829:
825:
813:
806:
796:
794:
790:
789:David Hilbert
786:
785:
780:
776:
771:
770:ideal numbers
767:
758:
757:for clarity.
756:
752:
748:
744:
739:
737:
736:number theory
733:
729:
725:
721:
720:prime numbers
717:
712:
708:
703:
701:
697:
693:
689:
688:quotient ring
685:
681:
677:
673:
670:is a special
669:
665:
661:
657:
645:
640:
638:
633:
631:
626:
625:
623:
622:
617:
616:
610:
606:
605:
604:
603:
602:
597:
596:
595:
590:
589:
588:
581:
577:
575:
571:
569:
565:
563:
562:Division ring
559:
558:
557:
556:
550:
545:
544:
516:
500:
498:
492:
476:
462:
461:-adic numbers
460:
455:
439:
425:
423:
418:
416:
412:
410:
403:
401:
397:
396:
395:
394:
393:
384:
380:
378:
374:
370:
366:
365:
364:
360:
358:
354:
352:
348:
346:
342:
340:
336:
334:
330:
329:
328:
324:
323:
322:
321:
315:
310:
309:
300:
296:
295:
294:
290:
286:
282:
280:
276:
275:
274:
270:
266:
262:
261:
260:
256:
255:
254:
253:
228:
224:
215:
212:
205:
204:Terminal ring
201:
178:
174:
173:
172:
168:
166:
162:
160:
156:
154:
150:
148:
144:
143:
142:
141:
140:
133:
129:
127:
123:
121:
117:
116:
115:
114:
113:
106:
102:
100:
96:
94:
90:
86:
82:
80:
76:
75:
74:
73:Quotient ring
70:
68:
64:
62:
58:
57:
56:
55:
46:
45:
42:
37:β Ring theory
36:
32:
31:
19:
14066:
14015:
13993:
13967:
13922:
13900:
13879:
13854:
13843:. Retrieved
13839:the original
13834:
13825:
13814:. Retrieved
13810:the original
13805:
13796:
13785:. Retrieved
13781:the original
13776:
13767:
13762:, p. 9.
13755:
13743:
13723:
13713:
13701:
13689:
13680:
13677:"Zero ideal"
13671:
13661:
13659:Lam (2001).
13653:
13641:
13629:
13598:
13586:
13574:
13562:
13550:
13541:
13535:
13526:
13520:
13511:
13483:
13479:
13475:
13471:
13465:
13464:) + ... + (β
13461:
13457:
13451:
13446:
13442:
13438:
13433:
13429:
13425:
13421:
13416:
13408:
13404:
13400:
13395:
13361:Ideal theory
13172:
12928:
12924:
12919:
12895:
12795:
12793:
12596:
12575:"; that is,
12505:
12436:
12294:, such that
12257:
12195:
11983:
11974:
11933:
11929:
11921:
11917:
11909:
11905:
11901:
11893:
11889:
11885:
11877:
11873:
11872:
11866:
11813:
11779:
11726:
11555:
11553:
11265:
11228:
11046:
11011:
10985:
10889:
10888:is prime in
10836:
10835:is prime in
10805:
10777:
10749:
10745:
10741:
10739:
10734:
10676:
10672:
10625:
10599:
10432:
10428:
10370:
10366:
10362:
10354:
10350:
10313:
10281:
10277:
10273:
10265:
10261:
10259:
10061:
9921:
9919:
9914:
9874:
9868:
9783:
9747:
9581:
9544:
9538:
9505:unit element
9473:
9358:
9173:
9169:
9147:
9087:
9040:
9034:
9026:
9024:
9019:
9014:
8995:
8205:
8120:
8025:
7904:
7774:
7611:
7575:
7555:
7554:
7479:
7383:
7156:
6950:
6917:
6889:
6885:
6883:
6640:
6543:
6489:
6477:
6473:
6469:
6464:
6460:
6455:
6451:
6447:
6442:
6438:
6432:
6426:
6422:
6418:
6414:
6410:
6403:
6397:
6392:
6378:
6374:
6370:
6366:
6362:
6355:
6351:
6321:
6233:
6232:is called a
6203:
6191:
6183:
6175:
6167:
6073:
6061:
6053:
6033:
6030:as a module.
6023:
6017:
6011:
6005:
5994:
5990:
5982:
5978:
5974:
5970:
5966:
5965:is called a
5956:
5951:reduced ring
5938:
5934:
5926:
5922:
5918:
5914:
5910:
5896:
5707:
5706:is called a
5681:
5655:
5626:: the ideal
5623:
5615:
5601:
5593:
5589:
5588:is called a
5579:
5573:factor rings
5570:
5566:
5565:
4997:
4993:
4802:
4798:
4650:
4646:
4640:
4611:
4580:
4576:
4572:
4568:
4452:
4448:
4413:
4347:Zorn's lemma
4339:
4335:
4184:
4177:
4027:
3971:
3967:
3929:
3815:-module and
3812:
3808:
3617:
3607:
3574:
3570:
3508:
3504:
3461:
3457:
2812:
2494:Take a ring
2146:. Likewise,
1974:
1928:
1781:(as it is a
1779:proper ideal
1778:
1741:
1736:
1641:
1637:
1633:
1629:
1623:
1607:
1418:
1413:. For a rng
1408:
1399:
1382:
1310:
1287:
1245:
1177:
1172:
1166:
1161:
1127:replaced by
1095:
1093:
1078:
874:"; that is,
814:is a subset
811:
802:
793:Emmy Noether
782:
766:Ernst Kummer
764:
754:
747:order theory
740:
716:prime ideals
704:
692:group theory
680:even numbers
663:
653:
613:
599:
598:
594:Free algebra
592:
591:
585:
584:
553:
496:
458:
421:
390:
389:
369:Finite field
318:
265:Finite field
251:
250:
177:Initial ring
137:
136:
110:
109:
66:
52:
18:Proper ideal
13964:Lang, Serge
13381:Ideal sheaf
12796:right ideal
12131:intersects
11938:prime ideal
11723:prime ideal
11319:shows that
10677:contraction
10353:(e.g. take
10182:. That is,
10064:. If (DCC)
9148:Indeed, if
7777:Tor functor
6385:. See also
6240:equals the
6040:annihilator
5969:if for all
5961:: An ideal
5917:if for any
5710:if for any
5708:prime ideal
5683:Prime ideal
5606:simple ring
4030:called the
3843:annihilator
3610:; see also
3178:matrix ring
3132:simple ring
2538:. For each
2421:polynomials
1640:called the
1614:Lie algebra
1169:commutative
1096:right ideal
1089:left module
799:Definitions
660:ring theory
656:mathematics
574:Simple ring
285:Jordan ring
159:Graded ring
41:Ring theory
14080:Categories
14046:0237.18005
13845:2017-01-14
13816:2017-01-14
13787:2017-01-14
13681:Math World
13494:References
13478:and every
13470:for every
13371:Ideal norm
12996:sub-module
12954:as a left
12686:and every
12648:For every
12597:left ideal
12506:left ideal
12401:lies over
11888:, and let
11560:surjective
10357:to be the
10272:, and let
9871:nilradical
9277:, meaning
7967:such that
6445:such that
5909:is called
5887:prime ring
5656:Unit ideal
5624:Zero ideal
5285:of a ring
4183:in a ring
3811:is a left
3056:such that
2940:such that
1933:skew-field
1897:for every
1742:zero ideal
1642:unit ideal
1632:, the set
1628:In a ring
1541:, we have
1477:and every
1451:for every
1419:left ideal
1389:surjective
998:and every
972:For every
822:that is a
812:left ideal
722:, and the
580:Commutator
339:GCD domain
13634:Lang 2005
13291:∈
13278:for some
12877:⊗
12865:∈
12859:⊗
12829:⊗
12817:∈
12811:⊗
12773:⊗
12739:⊗
12711:⊗
12699:∈
12671:⊗
12659:∈
12624:subobject
12530:subobject
12502:forgotten
12459:⊗
12273:−
12211:−
12144:−
12105:⇒
11823:⇔
11736:⇔
11584:
11578:⊇
11450:−
11420:−
11374:−
11365:−
11356:−
11301:±
11282:±
11212:−
11171:−
11067:→
11051:embedding
10964:⊆
10919:⊇
10846:⇒
10740:Assuming
10641:−
10575:∈
10552:∈
10507:∑
10371:extension
10359:inclusion
10145:
10124:⋅
10085:
10079:⊋
10060:for some
9991:
9973:
9939:
9888:
9845:
9839:⊂
9814:⋂
9798:
9692:⊊
9686:⊂
9629:⋅
9601:⊊
9518:−
9485:−
9450:∈
9432:−
9426:∣
9420:∈
9288:
9262:≃
9250:
9224:
9108:⋂
9065:
8999:Macaulay2
8967:≠
8923:∩
8872:∩
8138:∩
8086:
8068:∩
7924:⊂
7859:∩
7804:
7629:∩
7504:contains
7458:∩
7438:∩
7428:⊃
7402:∩
7050:∪
6863:…
6826:…
6795:∈
6768:∈
6755:∣
6729:⋯
6615:∈
6595:∈
6589:∣
6283:
6177:Nil ideal
6147:∈
6119:∈
6108:for some
6074:comaximal
6001:for some
5933:for some
5915:semiprime
5493:∼
5481:∈
5423:∼
5379:∼
5354:∈
5348:−
5320:∼
5228:…
5148:…
4964:∈
4945:∈
4926:∈
4905:∈
4899:∣
4863:⋯
4770:∈
4751:∈
4730:∈
4724:∣
4698:⋯
4564:for more.
4481:, taking
4466:≠
4397:⊂
4360:⊂
4302:∈
4295:⋃
4255:for each
4229:⊂
4161:∈
4128:idealizer
3910:∈
3889:∣
3883:∈
3862:
3826:⊂
3772:−
3764:↦
3727:↦
3639:→
3573:: unless
3477:−
3435:
3400:≠
3250:−
3230:
3200:→
3091:whenever
2818:The ring
2776:×
2748:≤
2742:≤
2690:×
2585:×
2557:≤
2551:≤
2260:The even
1910:∈
1878:∈
1864:−
1820:∈
1561:∈
1526:∈
1488:∈
1462:∈
1357:→
1228:∈
1222:−
1193:∼
1143:∈
1112:∈
1081:submodule
1011:∈
983:∈
779:Dirichlet
521:∞
299:Semifield
14014:(1971).
13966:(2005).
13921:(1995),
13878:(1969).
13773:"ideals"
13721:(1987).
13445:+ ... +
13340:See also
12476:, where
10808:, then:
10744: :
10276: :
9361:-module
8381:. Then,
8261:and let
8048:we have
7571:quantale
7560:complete
6020:element.
5513:. Then
5371:. Then
4187:; i.e.,
3424:), then
2601:matrices
2262:integers
2093:; i.e.,
1289:quotient
931:subgroup
824:subgroup
803:Given a
781:'s book
676:integers
293:Semiring
279:Lie ring
61:Subrings
14038:0349811
13957:1322960
13334:
13312:
13308:
13280:
13253:, i.e.
13251:
13229:
13203:
13175:
13168:
13148:
13104:
13084:
13020:
13000:
12932:-module
12892:
12848:
12844:
12800:
12788:
12756:
12726:
12688:
12573:
12553:
12474:
12442:
12427:
12403:
12375:
12332:
12292:
12262:
12230:
12200:
12163:
12133:
12054:
12010:
11924:be the
11896:be the
11693:
11649:
11549:
11491:
11487:
11406:
11402:
11321:
11095:
11054:
10628:, then
10470:
10437:
10369:). The
10316:, then
10268:be two
10250:
10184:
10012:
9962:
9928:, then
9778:
9752:
9707:
9666:
9664:and so
9616:
9590:
9580:, then
9578:
9549:
9395:
9363:
9176:, then
9043:-module
9039:simple
8379:
8263:
8201:
8181:
8012:
7969:
7900:
7781:
7769:
7745:
7743:modulo
7614:, then
7378:
7297:
7291:
7210:
7203:
7159:
7152:
7128:
7069:
7035:
6946:
6922:
6540:
6516:
6316:unmixed
6312:
6246:
6236:if its
6162:
6136:
5937:, then
5911:radical
5883:
5863:
5819:
5799:
5772:
5752:
5555:
5535:
5511:
5461:
5457:
5437:
5413:
5393:
5369:
5337:
5307:
5287:
5256:
5204:
5116:
5090:
5060:
5025:
5023:(resp.
4608:
4585:
4558:
4526:
4283:
4257:
4124:
4084:
3798:
3753:
3654:
3622:
3511:, then
3422:
3382:
3358:
3309:
3305:
3285:
3174:
3136:
3125:
3093:
2977:
2942:
2849:of all
2763:
2731:
2727:
2707:
2679:of all
2572:
2540:
2536:
2516:
2489:
2458:
2362:
2337:
2313:
2288:
2195:. Then
2144:
2124:
1935:, then
1925:
1899:
1772:
1746:
1699:
1672:
1158:
1129:
1073:
1053:
1026:
1000:
967:
935:
872:
852:
826:of the
761:History
684:closure
495:PrΓΌfer
97:β’
14044:
14036:
14026:
14000:
13978:
13955:
13945:
13907:
13888:
13731:
13460:) + (β
13450:, and
12754:is in
12498:monoid
11912:is an
11876:: Let
11874:Remark
11600:then:
9924:is an
9045:. The
8908:while
8121:since
7556:Remark
6048:module
6044:simple
5997:is in
5985:is in
5941:is in
5929:is in
5861:is in
5797:is in
5459:, let
5309:, let
4176:be an
3970:, the
3614:below.
2916:under
2813:column
2623:whose
1579:i.e.,
1447:is in
1396:kernel
1246:is an
1051:is in
672:subset
147:Module
120:Kernel
13387:Notes
12920:ideal
12894:". A
12622:is a
12595:is a
12528:is a
11928:from
11880:be a
11808:is a
11721:is a
11231:. So
11097:. In
10288:. If
10284:be a
9503:is a
6951:Note
6888:with
6467:with
6256:grade
6238:grade
6046:left
6042:of a
5981:, if
5925:, if
5774:, if
5610:field
5596:with
5118:. If
4649:over
3215:is a
2872:from
2412:is a
2390:, so
1931:is a
1511:, if
1173:ideal
929:is a
743:ideal
666:of a
664:ideal
662:, an
499:-ring
363:Field
259:Field
67:Ideal
54:Rings
14024:ISBN
13998:ISBN
13976:ISBN
13943:ISBN
13905:ISBN
13886:ISBN
13729:ISBN
13403:the
12504:. A
11904:and
11892:and
11647:and
11562:and
10264:and
10260:Let
9746:and
9205:and
9025:Let
8490:and
8206:Let
8179:and
7126:and
7009:and
6916:and
6514:and
6463:and
6275:proj
6134:and
5993:and
5973:and
5841:and
5730:and
4524:and
4268:<
4133:Let
3110:>
3041:>
2811:-th
1417:, a
1160:. A
810:, a
805:ring
694:, a
668:ring
14042:Zbl
13935:doi
13474:in
13420:If
13407:of
13106:.
12998:of
12626:of
12508:of
12256:of
11973:of
11932:to
11916:of
11900:of
11884:of
11812:in
11725:in
11581:ker
11558:is
10733:to
10709:of
10600:If
10427:in
10403:of
10142:Ann
10082:Ann
9988:Jac
9970:nil
9936:Jac
9920:If
9913:of
9885:nil
9873:of
9842:Jac
9795:nil
9717:If
9285:Ann
9247:Ann
9221:Ann
9086:of
9062:Jac
9033:of
8083:lcm
8026:In
7902:.)
7790:Tor
7576:If
7528:or
6920:in
6892:in
6381:is
6365:of
6280:dim
6076:if
6018:one
5977:in
5921:in
5913:or
5901:or
5750:in
5435:on
5335:if
4571:of
4058:by
4034:of
3928:of
3853:Ann
3569:of
3460:of
3432:ker
3227:ker
3183:If
2894:to
1411:rng
1308:.)
1296:by
1292:of
1250:on
1178:If
1083:of
933:of
830:of
818:of
745:in
738:).
654:In
14082::
14065:.
14040:.
14034:MR
14032:.
14022:.
13974:.
13953:MR
13951:,
13941:,
13933:,
13925:,
13874:;
13833:.
13804:.
13775:.
13679:.
13610:^
13502:^
13456:(β
13441:+
13336:.
13170:.
12794:A
11981:.
11551:.
11404:,
10780:,
10748:β
10737:.
10280:β
9917:.
9618:,
9001:.
8203:.
8018:.
7779::
7573:.
7552:.
7205:,
7154:.
6948:.
6886:ab
6700::=
6574::=
6476:=
6474:BA
6472:=
6470:AB
6454:=
6452:BA
6450:=
6448:AB
6068:,
5983:ab
5678:).
4653::
2257:.)
1616:.
1605:.
1094:A
795:.
702:.
607:β’
578:β’
572:β’
566:β’
560:β’
493:β’
456:β’
419:β’
413:β’
404:β’
398:β’
381:β’
375:β’
367:β’
361:β’
355:β’
349:β’
343:β’
337:β’
331:β’
325:β’
297:β’
291:β’
283:β’
277:β’
271:β’
263:β’
257:β’
202:β’
175:β’
169:β’
163:β’
157:β’
151:β’
145:β’
130:β’
124:β’
118:β’
103:β’
91:β’
83:β’
77:β’
71:β’
65:β’
59:β’
14071:.
14048:.
14006:.
13984:.
13937::
13913:.
13894:.
13848:.
13819:.
13790:.
13737:.
13484:R
13480:n
13476:X
13472:x
13468:)
13466:x
13462:x
13458:x
13452:n
13447:x
13443:x
13439:x
13434:n
13430:R
13426:X
13422:R
13411:.
13409:R
13401:R
13321:Z
13295:Z
13288:m
13265:Z
13261:m
13238:Z
13214:Z
13190:Z
13186:=
13183:R
13156:R
13134:R
13114:I
13092:R
13070:R
13050:R
13030:I
13008:R
12982:I
12962:R
12942:R
12929:R
12906:R
12880:)
12874:,
12871:I
12868:(
12862:r
12856:x
12832:)
12826:,
12823:I
12820:(
12814:x
12808:r
12790:.
12776:)
12770:,
12767:I
12764:(
12742:x
12736:r
12714:)
12708:,
12705:I
12702:(
12696:x
12674:)
12668:,
12665:R
12662:(
12656:r
12634:R
12610:I
12583:I
12561:R
12539:I
12516:R
12484:R
12462:)
12456:,
12453:R
12450:(
12413:p
12387:q
12360:p
12355:B
12349:e
12343:p
12315:p
12310:B
12304:q
12278:p
12270:A
12258:B
12242:q
12216:p
12208:A
12196:B
12179:p
12174:B
12149:p
12141:A
12117:e
12111:p
12099:p
12094:B
12090:=
12084:p
12079:B
12073:e
12067:p
12040:c
12037:e
12031:p
12025:=
12020:p
11994:p
11975:A
11959:p
11954:=
11949:a
11934:B
11930:A
11922:f
11918:A
11910:B
11906:L
11902:K
11894:A
11890:B
11886:L
11878:K
11869:.
11867:B
11851:e
11845:a
11814:A
11794:a
11782:.
11780:B
11764:e
11758:a
11727:A
11707:a
11695:.
11679:b
11674:=
11669:e
11666:c
11660:b
11633:a
11628:=
11623:c
11620:e
11614:a
11587:f
11573:a
11556:f
11535:2
11531:)
11527:i
11524:+
11521:1
11518:(
11515:=
11510:e
11506:)
11502:2
11499:(
11475:)
11470:2
11466:)
11462:i
11459:+
11456:1
11453:(
11447:)
11444:i
11441:+
11438:1
11435:(
11432:(
11429:=
11426:)
11423:i
11417:1
11414:(
11390:)
11385:2
11381:)
11377:i
11371:1
11368:(
11362:)
11359:i
11353:1
11350:(
11347:(
11344:=
11341:)
11338:i
11335:+
11332:1
11329:(
11307:i
11304:2
11298:=
11293:2
11289:)
11285:i
11279:1
11276:(
11266:B
11250:e
11246:)
11242:2
11239:(
11229:B
11215:i
11209:1
11206:,
11203:i
11200:+
11197:1
11177:)
11174:i
11168:1
11165:(
11162:)
11159:i
11156:+
11153:1
11150:(
11147:=
11144:2
11123:]
11120:i
11117:[
11112:Z
11108:=
11105:B
11082:]
11079:i
11076:[
11071:Z
11063:Z
11047:B
11031:e
11025:a
11012:A
10996:a
10969:b
10959:e
10956:c
10950:b
10924:a
10914:c
10911:e
10905:a
10892:.
10890:A
10874:c
10868:b
10837:B
10821:b
10806:B
10790:b
10778:A
10762:a
10750:B
10746:A
10742:f
10735:A
10719:b
10695:c
10689:b
10673:A
10659:)
10654:b
10649:(
10644:1
10637:f
10626:B
10610:b
10583:}
10578:B
10570:i
10566:y
10562:,
10557:a
10547:i
10543:x
10539::
10536:)
10531:i
10527:x
10523:(
10520:f
10515:i
10511:y
10502:{
10497:=
10492:e
10486:a
10458:)
10453:a
10448:(
10445:f
10433:B
10429:B
10413:a
10389:e
10383:a
10367:Q
10363:Z
10355:f
10351:B
10337:)
10332:a
10327:(
10324:f
10314:A
10298:a
10282:B
10278:A
10274:f
10266:B
10262:A
10238:0
10235:=
10230:a
10223:1
10220:+
10217:n
10213:J
10209:=
10204:a
10197:n
10193:J
10170:0
10167:=
10164:)
10161:)
10156:n
10152:J
10148:(
10138:/
10132:a
10127:(
10121:J
10101:)
10096:n
10092:J
10088:(
10074:a
10062:n
10046:1
10043:+
10040:n
10036:J
10032:=
10027:n
10023:J
10000:)
9997:R
9994:(
9985:=
9982:)
9979:R
9976:(
9948:)
9945:R
9942:(
9922:R
9915:R
9897:)
9894:R
9891:(
9875:R
9854:)
9851:R
9848:(
9834:p
9820:p
9810:=
9807:)
9804:R
9801:(
9780:.
9766:0
9763:=
9760:M
9748:M
9734:M
9731:=
9728:M
9725:J
9695:M
9689:L
9683:M
9680:J
9677:=
9674:M
9652:0
9649:=
9646:)
9643:L
9639:/
9635:M
9632:(
9626:J
9604:M
9598:L
9582:M
9566:M
9563:=
9560:M
9557:J
9545:M
9524:y
9521:x
9515:1
9491:x
9488:y
9482:1
9459:.
9456:}
9453:R
9447:y
9438:x
9435:y
9429:1
9423:R
9417:x
9414:{
9411:=
9408:J
9381:m
9375:/
9371:R
9359:R
9343:m
9319:m
9297:)
9294:M
9291:(
9265:M
9259:)
9256:x
9253:(
9243:/
9239:R
9236:=
9233:)
9230:M
9227:(
9217:/
9213:R
9193:M
9190:=
9187:x
9184:R
9174:M
9170:x
9156:M
9133:.
9128:m
9114:m
9104:=
9101:J
9088:R
9074:)
9071:R
9068:(
9059:=
9056:J
9041:R
9035:R
9027:R
8979:c
8972:a
8964:)
8959:2
8955:z
8951:+
8948:z
8945:x
8942:,
8939:w
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8933:=
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8918:a
8894:b
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8882:=
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8831:,
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8822:+
8819:w
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8813:,
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8799:2
8795:z
8791:+
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8785:x
8782:(
8779:=
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8683:z
8680:x
8677:+
8672:2
8668:z
8664:(
8661:=
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8655:)
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8643:(
8640:w
8637:,
8634:)
8631:z
8628:+
8625:x
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8601:(
8598:z
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8589:z
8586:+
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8580:(
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8574:(
8571:=
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8533:x
8530:,
8527:w
8524:,
8521:z
8518:(
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8430:+
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8312:x
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8225:[
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8217:=
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8101:)
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7998:c
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7986:=
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7929:b
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7873:/
7869:)
7864:b
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7825:,
7820:a
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7771:.
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7699:1
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7405:(
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7380:.
7364:c
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7335:=
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7320:b
7315:+
7310:a
7305:(
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7277:c
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7260:b
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7248:=
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7220:a
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6902:a
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6860:,
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6848:=
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6835:;
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6823:,
6820:2
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6811:=
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6318:.
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6264:I
6261:(
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6200:.
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6150:J
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6116:x
6096:1
6093:=
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5496:0
5490:x
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5475:{
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5101:x
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5073:x
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4973:.
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4803:X
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4779:.
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4765:i
4761:x
4757:,
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4715:x
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4672:{
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4625:X
4622:R
4612:X
4596:X
4593:R
4581:X
4577:X
4573:R
4569:X
4546:}
4543:1
4540:{
4537:=
4534:E
4512:)
4509:0
4506:(
4503:=
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4449:E
4433:0
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4392:0
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4097:a
4092:(
4068:b
4044:a
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3997:a
3992:+
3987:b
3982:(
3972:R
3968:R
3952:b
3947:,
3942:a
3930:S
3916:}
3913:S
3907:s
3904:,
3901:0
3898:=
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3892:r
3886:R
3880:r
3877:{
3874:=
3871:)
3868:S
3865:(
3857:R
3829:M
3823:S
3813:R
3809:M
3786:)
3783:J
3780:(
3775:1
3768:f
3761:J
3739:)
3736:I
3733:(
3730:f
3724:I
3704:S
3684:f
3664:R
3642:S
3636:R
3633::
3630:f
3608:S
3594:)
3591:I
3588:(
3585:f
3575:f
3571:S
3557:)
3554:R
3551:(
3548:f
3528:)
3525:I
3522:(
3519:f
3509:R
3505:I
3491:)
3488:I
3485:(
3480:1
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3458:I
3444:)
3441:f
3438:(
3408:S
3404:0
3395:S
3391:1
3368:S
3344:S
3340:1
3336:=
3333:)
3328:R
3324:1
3320:(
3317:f
3293:R
3271:)
3266:S
3262:0
3258:(
3253:1
3246:f
3242:=
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3236:f
3233:(
3203:S
3197:R
3194::
3191:f
3162:)
3159:1
3156:(
3153:,
3150:)
3147:0
3144:(
3127:.
3113:L
3107:|
3104:x
3101:|
3079:0
3076:=
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3070:x
3067:(
3064:f
3044:0
3038:L
3018:f
2998:)
2994:R
2990:(
2987:C
2965:0
2962:=
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2956:1
2953:(
2950:f
2928:f
2903:R
2881:R
2860:f
2837:)
2833:R
2829:(
2826:C
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2631:i
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2582:n
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2548:1
2524:n
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2491:.
2477:]
2474:x
2471:[
2467:R
2444:1
2441:+
2436:2
2432:x
2416:.
2399:Z
2373:Z
2349:Z
2345:n
2323:n
2300:Z
2296:2
2273:Z
2245:x
2242:=
2239:x
2236:)
2233:y
2230:z
2227:(
2224:=
2221:)
2218:x
2215:y
2212:(
2209:z
2206:=
2203:z
2183:z
2163:1
2160:=
2157:y
2154:z
2132:y
2110:1
2107:=
2104:x
2101:y
2081:)
2078:1
2075:(
2072:=
2069:x
2066:R
2046:x
2043:R
2023:x
2003:)
2000:1
1997:(
1994:,
1991:)
1988:0
1985:(
1975:R
1961:)
1958:1
1955:(
1952:,
1949:)
1946:0
1943:(
1929:R
1913:R
1907:r
1883:a
1875:u
1872:)
1867:1
1860:u
1856:r
1853:(
1850:=
1847:r
1825:a
1817:u
1795:a
1760:)
1757:0
1754:(
1737:R
1722:}
1717:R
1713:0
1709:{
1685:R
1681:1
1658:)
1655:1
1652:(
1638:R
1634:R
1630:R
1593:R
1590:=
1587:I
1567:;
1564:I
1558:1
1555:r
1552:=
1549:r
1529:R
1523:r
1513:I
1509:R
1505:I
1491:I
1485:x
1465:R
1459:r
1449:I
1435:x
1432:r
1422:I
1415:R
1385:R
1368:I
1364:/
1360:R
1354:R
1331:I
1327:/
1323:R
1313:I
1298:I
1294:R
1274:I
1270:/
1266:R
1252:R
1231:I
1225:y
1219:x
1196:y
1190:x
1180:I
1146:I
1140:r
1137:x
1115:I
1109:x
1106:r
1085:R
1075:.
1061:I
1039:x
1036:r
1014:I
1008:x
986:R
980:r
969:,
955:)
952:+
949:,
946:R
943:(
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914:+
911:,
908:I
905:(
882:I
860:R
838:R
820:R
816:I
808:R
643:e
636:t
629:v
526:)
517:p
513:(
509:Z
497:p
477:p
472:Q
459:p
440:p
435:Z
422:p
408:n
233:Z
229:1
225:/
220:Z
216:=
213:0
187:Z
20:)
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