2936:
1873:
1756:
104:
2834:
2897:
1283:
is left
Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if it is right Artinian. The analogous statements with "right" and "left" interchanged are also true.
2370:
1498:
2740:
2311:
948:
739:
165:
677:
1868:{\displaystyle R=\left\{\left.{\begin{bmatrix}a&\beta \\0&\gamma \end{bmatrix}}\,\right\vert \,a\in \mathbf {Z} ,\beta \in \mathbf {Q} ,\gamma \in \mathbf {Q} \right\}.}
823:
3123:
3091:
3067:
3039:
1454:
1703:
Indeed, there are rings that are right
Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. For example, if
975:
874:
2922:, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary.
2597:
2574:
2526:
2506:
2482:
2462:
2442:
2413:
2393:
2248:
2224:
2192:
2159:
2139:
2115:
2095:
2067:
2044:
2021:
1111:
1048:
847:
610:, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa.
3588:
3164:
says that, over a left
Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the
558:
1969:
50:
2745:
3360:. Mathematics and Its Applications. Soviet Series. Vol. 70. Translated by Bakhturin, Yu. A. Dordrecht: Kluwer Academic Publishers.
2850:
1309:(left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of
2918:, already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and (Noetherian)
3742:
3712:
3636:
3539:
3519:
3373:
2317:
1467:
1663:
is a subring of a field, any integral domain that is not
Noetherian provides an example. To give a less trivial example,
169:
Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is
2675:
2915:
551:
1652:
1519:
Rings that are not
Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings:
3569:
3498:
1593:
is not
Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (2), (2), (2), ...
1246:
3583:
1972:. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain.
2262:
1167:
882:
1979:
is not
Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in
3681:
2230:
This is because there is a bijection between the left and right ideals of the group ring in this case, via the
3768:
3482:
686:
544:
115:
1276:
3763:
2847:, on the other hand, gives some information about a descending chain of ideals given by powers of ideals
1965:
1360:
636:
630:
413:
224:
1392:, such as the integers, is Noetherian since every ideal is generated by a single element. This includes
782:
3734:
3094:
2911:
2904:
1325:
1051:
220:
205:
201:
193:
2947:
39:; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said
3758:
3514:
Atiyah, M. F., MacDonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley-Longman.
3235:
3165:
3104:
3072:
3048:
3020:
2919:
2625:
1435:
32:
1926:
1203:
1170:
over a commutative
Noetherian ring is Noetherian. (This follows from the two previous properties.)
184:
ring theory since many rings that are encountered in mathematics are
Noetherian (in particular the
170:
504:
200:), and many general theorems on rings rely heavily on the Noetherian property (for example, the
3668:. Graduate Texts in Mathematics. Vol. 131 (2nd ed.). New York: Springer. p. 19.
1457:
1393:
1269:
1055:
1515:
The ring of polynomials in finitely-many variables over the integers or a field is
Noetherian.
3008:
2914:
of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem,
2621:
1509:
1310:
1302:
3785:
3691:
3646:
3508:
3391:
3161:
2844:
1389:
1344:
1291:
1182:
953:
852:
491:
483:
455:
450:
441:
398:
340:
3722:
3654:
3399:
8:
2967:
2837:
2633:
2251:
2074:
2024:
1597:
1336:
1321:
1153:
1121:
577:
509:
499:
350:
250:
242:
233:
181:
36:
1002:
3607:
3579:
2900:
2582:
2550:
2511:
2491:
2467:
2447:
2418:
2398:
2378:
2233:
2200:
2168:
2144:
2124:
2100:
2080:
2070:
2052:
2047:
2029:
1997:
1980:
1352:
1061:
1024:
877:
832:
315:
306:
264:
28:
3738:
3708:
3677:
3632:
3565:
3535:
3515:
3494:
3379:
3369:
3252:
3177:
3153:
2982:
over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring
1668:
1590:
1189:
1160:
753:
3718:
3669:
3650:
3624:
3597:
3557:
3525:
3486:
3395:
3361:
3244:
2979:
2608:
2599:
whose group ring over any Noetherian commutative ring is not two-sided Noetherian.
2577:
2540:
2533:
2485:
2118:
1411:) is a Noetherian domain in which every ideal is generated by at most two elements.
1408:
1397:
1306:
607:
335:
185:
177:
3248:
3230:
360:
3687:
3642:
3529:
3504:
3387:
3015:
2529:
1660:
1415:
1404:
1374:
1196:
1019:
998:
989:
of the ring is finitely generated. However, it is not enough to ask that all the
757:
427:
421:
408:
388:
379:
345:
282:
189:
1659:
However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any
3549:
2536:
1976:
1419:
1382:
1178:
469:
3673:
3628:
3561:
3490:
3365:
1968:
is not necessarily a Noetherian ring. It does satisfy a weaker condition: the
1456:
is a both left and right Noetherian ring; this follows from the fact that the
3779:
3661:
3383:
3256:
3182:
2629:
1287:
1280:
1140:
990:
772:
355:
320:
277:
216:
3332:
2641:
2544:
2254:
1505:
1265:
is a Noetherian ring (see the "faithfully flat" article for the reasoning).
1258:
529:
460:
294:
212:
197:
106:
of left (or right) ideals has a largest element; that is, there exists an
3226:
3098:
2843:
A Noetherian ring is defined in terms of ascending chains of ideals. The
1501:
1430:
1378:
986:
519:
514:
403:
393:
367:
20:
3485:, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376,
2935:
3700:
3611:
3157:
1992:
1715:
1295:
1157:
994:
269:
1385:, is Noetherian. (A field only has two ideals — itself and (0).)
580:, it is necessary to distinguish between three very similar concepts:
3141:
745:
524:
330:
287:
255:
3602:
1422:
is a Noetherian ring, as a consequence of the Hilbert basis theorem.
99:{\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots }
2978:
Given a ring, there is a close connection between the behaviors of
2829:{\displaystyle (f)=(p_{1}^{n_{1}})\cap \cdots \cap (p_{r}^{n_{r}})}
1937:
of the previous paragraph is a subring of the left Noetherian ring
1878:
This ring is right Noetherian, but not left Noetherian; the subset
1708:
767:
The following condition is also an equivalent condition for a ring
325:
1600:
from the real numbers to the real numbers is not Noetherian: Let
3556:. Graduate Texts in Mathematics. Vol. 150. Springer-Verlag.
2836:
and thus the primary decomposition is a direct generalization of
2607:
Many important theorems in ring theory (especially the theory of
1211:
173:. A ring is Noetherian if it is both left- and right-Noetherian.
2892:{\displaystyle I\supseteq I^{2}\supseteq I^{3}\supseteq \cdots }
2576:
is two-sided Noetherian. On the other hand, however, there is a
3333:
The ring of stable homotopy groups of spheres is not noetherian
1320:
In a commutative Noetherian ring, there are only finitely many
985:
For a commutative ring to be Noetherian it suffices that every
259:
3619:
Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki (2008),
1301:(Bass) A ring is (left/right) Noetherian if and only if every
1199:
Noetherian module over it, then the ring is a Noetherian ring.
595:
if it satisfies the ascending chain condition on right ideals.
215:, but the importance of the concept was recognized earlier by
1894:= 0 is a left ideal that is not finitely generated as a left
1348:
588:
if it satisfies the ascending chain condition on left ideals.
3733:, Cambridge Studies in Advanced Mathematics (2nd ed.),
1772:
3554:
Commutative Algebra with a View Toward Algebraic Geometry
2611:) rely on the assumptions that the rings are Noetherian.
1001:(see a counterexample to Krull's intersection theorem at
223:(which asserts that polynomial rings are Noetherian) and
2742:
is a product of powers of distinct prime elements, then
2365:{\displaystyle g\mapsto g^{-1}\qquad (\forall g\in G).}
3623:, Progress in Mathematics, vol. 236, BirkhÀuser,
3621:
D-modules, perverse sheaves, and representation theory
3311:, Ch III, §2, no. 10, Remarks at the end of the number
1780:
1648:, etc., is an ascending chain that does not terminate.
1523:
The ring of polynomials in infinitely-many variables,
1347:). Thus, if, in addition, the factorization is unique
885:
3618:
3320:
3231:"Commutative rings with restricted minimum condition"
3107:
3075:
3051:
3023:
2853:
2748:
2678:
2620:
Over a commutative Noetherian ring, each ideal has a
2585:
2553:
2514:
2494:
2470:
2450:
2421:
2401:
2381:
2320:
2265:
2236:
2203:
2171:
2147:
2127:
2103:
2083:
2055:
2032:
2000:
1759:
1493:{\displaystyle \operatorname {Sym} ({\mathfrak {g}})}
1470:
1438:
1314:
1064:
1027:
993:
are finitely generated, as there is a non-Noetherian
956:
855:
835:
785:
689:
639:
118:
53:
613:
There are other, equivalent, definitions for a ring
3578:
3343:
2973:
2735:{\displaystyle f=p_{1}^{n_{1}}\cdots p_{r}^{n_{r}}}
1905:is a commutative subring of a left Noetherian ring
3707:(Third ed.), Reading, Mass.: Addison-Wesley,
3117:
3085:
3061:
3033:
2891:
2828:
2734:
2591:
2568:
2520:
2500:
2476:
2456:
2436:
2407:
2387:
2364:
2305:
2242:
2218:
2186:
2153:
2133:
2109:
2089:
2061:
2038:
2015:
1867:
1746:. Choosing a basis, we can describe the same ring
1492:
1448:
1105:
1042:
969:
942:
868:
841:
817:
733:
671:
159:
98:
764:Similar results hold for right-Noetherian rings.
47:respectively. That is, every increasing sequence
3777:
3589:Proceedings of the American Mathematical Society
3292:
3290:
2161:, the following two conditions are equivalent.
1500:, which is a polynomial ring over a field (the
1272:of a commutative Noetherian ring is Noetherian.
3476:
3460:
3448:
3436:
3424:
3296:
1504:); thus, Noetherian. For the same reason, the
1237:is a subring of a commutative Noetherian ring
3287:
2306:{\displaystyle R\to R^{\operatorname {op} },}
1970:ascending chain condition on principal ideals
1586:), etc. is ascending, and does not terminate.
552:
3477:Anderson, Frank W.; Fuller, Kent R. (1992),
1152:is also Noetherian. Stated differently, the
943:{\textstyle f_{i}=\sum _{j=1}^{n}r_{j}f_{j}}
3355:
3349:
3041:such that each injective left module over
559:
545:
16:Mathematical ring with well-behaved ideals
3728:
3601:
3356:OlâshanskiÄ, Aleksandr Yurâevich (1991).
3281:
3269:
3156:of an indecomposable injective module is
2996:(Bass) Each direct sum of injective left
2899:. It is a technical tool that is used to
2444:is left/right/two-sided Noetherian, then
1986:
1921:is Noetherian. (In the special case when
1817:
1811:
1607:be the ideal of all continuous functions
602:if it is both left- and right-Noetherian.
176:Noetherian rings are fundamental in both
3548:
3524:
3412:
3358:Geometry of defining relations in groups
3308:
3213:
2926:
2624:, meaning that it can be written as an
2464:is left/right/two-sided Noetherian and
1335:, every element can be factorized into
734:{\displaystyle I=Ra_{1}+\cdots +Ra_{n}}
3778:
3666:A first course in noncommutative rings
160:{\displaystyle I_{n}=I_{n+1}=\cdots .}
3225:
2508:is a Noetherian commutative ring and
3699:
3582:; Jategaonkar, Arun Vinayak (1974).
3321:Hotta, Takeuchi & Tanisaki (2008
3200:
3198:
2930:
1222:is a finitely generated module over
571:
3660:
3110:
3078:
3054:
3026:
2614:
1482:
1441:
1373:Any field, including the fields of
1331:In a commutative Noetherian domain
1163:of a Noetherian ring is Noetherian.
672:{\displaystyle a_{1},\ldots ,a_{n}}
13:
2344:
818:{\displaystyle f_{1},f_{2},\dots }
14:
3797:
3751:
3531:Commutative Algebra: Chapters 1-7
3195:
2636:are all distinct) where an ideal
1925:is commutative, this is known as
1653:stable homotopy groups of spheres
1351:multiplication of the factors by
1315:#Implication on injective modules
1214:of a commutative Noetherian ring
211:Noetherian rings are named after
3069:-generated modules (a module is
2986:, the following are equivalent:
2974:Implication on injective modules
2934:
1953:is finitely generated as a left
1929:.) However, this is not true if
1913:is finitely generated as a left
1853:
1839:
1825:
1544:, etc. The sequence of ideals (
1277:AkizukiâHopkinsâLevitzki theorem
771:to be left-Noetherian and it is
3479:Rings and categories of modules
3454:
3442:
3430:
3418:
3406:
3344:Formanek & Jategaonkar 1974
3337:
3144:into a direct sum of copies of
3118:{\displaystyle {\mathfrak {c}}}
3086:{\displaystyle {\mathfrak {c}}}
3062:{\displaystyle {\mathfrak {c}}}
3034:{\displaystyle {\mathfrak {c}}}
2916:Krull's principal ideal theorem
2903:other key theorems such as the
2602:
2340:
1449:{\displaystyle {\mathfrak {g}}}
1286:A left Noetherian ring is left
1195:If a commutative ring admits a
1139:is a two-sided ideal, then the
1018:is a Noetherian ring, then the
3584:"Subrings of Noetherian rings"
3326:
3314:
3302:
3275:
3263:
3219:
3207:
3014:(FaithâWalker) There exists a
2823:
2798:
2786:
2761:
2755:
2749:
2563:
2557:
2431:
2425:
2356:
2341:
2324:
2291:
2284:
2278:
2275:
2269:
2213:
2207:
2181:
2175:
2010:
2004:
1487:
1477:
1181:every finitely generated left
1100:
1068:
1037:
1031:
1:
3483:Graduate Texts in Mathematics
3470:
3249:10.1215/S0012-7094-50-01704-2
2672:. For example, if an element
1933:is not commutative: the ring
1726:be the ring of homomorphisms
1008:
3729:Matsumura, Hideyuki (1989),
2840:of integers and polynomials.
1886:consisting of elements with
1508:, and more general rings of
1313:injective modules. See also
1253:(or more generally exhibits
1113:is a Noetherian ring. Also,
633:, i.e. there exist elements
7:
3764:Encyclopedia of Mathematics
3171:
3007:-module is a direct sum of
2640:is called primary if it is
1966:unique factorization domain
1367:
1361:unique factorization domain
1003:Local ring#Commutative case
194:rings of algebraic integers
10:
3802:
3735:Cambridge University Press
3461:Anderson & Fuller 1992
3449:Anderson & Fuller 1992
3437:Anderson & Fuller 1992
3425:Anderson & Fuller 1992
3323:, §D.1, Proposition 1.4.6)
3297:Anderson & Fuller 1992
2993:is a left Noetherian ring.
2920:universally catenary rings
2905:Krull intersection theorem
2668:for some positive integer
1687:is a subring of the field
1627:. The sequence of ideals
1326:descending chain condition
829:, there exists an integer
206:Krull intersection theorem
3674:10.1007/978-1-4419-8616-0
3629:10.1007/978-0-8176-4523-6
3562:10.1007/978-1-4612-5350-1
3491:10.1007/978-1-4612-4418-9
3366:10.1007/978-94-011-3618-1
3236:Duke Mathematical Journal
1166:Every finitely-generated
1133:is a Noetherian ring and
775:'s original formulation:
33:ascending chain condition
3188:
1961:is not left Noetherian.
1699:) in only two variables.
1429:of a finite-dimensional
225:Hilbert's syzygy theorem
3731:Commutative Ring Theory
3093:-generated if it has a
2141:and a commutative ring
1983:but is not Noetherian.
1425:The enveloping algebra
1394:principal ideal domains
1124:, is a Noetherian ring.
1052:Hilbert's basis theorem
997:whose maximal ideal is
617:to be left-Noetherian:
221:Hilbert's basis theorem
3119:
3087:
3063:
3035:
3000:-modules is injective.
2893:
2830:
2736:
2593:
2570:
2522:
2502:
2478:
2458:
2438:
2409:
2389:
2366:
2307:
2244:
2220:
2188:
2155:
2135:
2111:
2091:
2063:
2040:
2017:
1987:Noetherian group rings
1869:
1510:differential operators
1494:
1458:associated graded ring
1450:
1328:holds on prime ideals.
1290:and a left Noetherian
1107:
1044:
971:
944:
919:
870:
843:
819:
748:set of left ideals of
735:
673:
202:LaskerâNoether theorem
161:
100:
3166:KrullâSchmidt theorem
3136:such that every left
3120:
3088:
3064:
3036:
2894:
2831:
2737:
2622:primary decomposition
2594:
2571:
2523:
2503:
2479:
2459:
2439:
2410:
2390:
2367:
2308:
2245:
2221:
2189:
2156:
2136:
2112:
2092:
2064:
2041:
2018:
1870:
1734:to itself satisfying
1495:
1451:
1275:A consequence of the
1233:Similarly, if a ring
1230:is a Noetherian ring.
1108:
1050:is Noetherian by the
1045:
972:
970:{\displaystyle r_{j}}
945:
899:
871:
869:{\displaystyle f_{i}}
844:
820:
736:
674:
162:
101:
3299:, Proposition 18.13.
3128:There exists a left
3105:
3073:
3049:
3021:
3003:Each injective left
2927:Non-commutative case
2851:
2746:
2676:
2583:
2551:
2512:
2492:
2468:
2448:
2419:
2399:
2379:
2318:
2263:
2234:
2226:is right-Noetherian.
2201:
2169:
2145:
2125:
2101:
2081:
2053:
2030:
1998:
1757:
1598:continuous functions
1468:
1436:
1390:principal ideal ring
1345:factorization domain
1337:irreducible elements
1322:minimal prime ideals
1062:
1025:
954:
883:
853:
833:
783:
756:by inclusion, has a
687:
637:
578:noncommutative rings
456:Group with operators
399:Complemented lattice
234:Algebraic structures
219:, with the proof of
116:
51:
3534:. Springer-Verlag.
3427:, Theorem 25.6. (b)
3415:, Proposition 3.11.
3045:is a direct sum of
2838:prime factorization
2822:
2785:
2731:
2706:
2252:associative algebra
2194:is left-Noetherian.
2075:associative algebra
1279:is that every left
1177:is left-Noetherian
1168:commutative algebra
510:Composition algebra
270:Quasigroup and loop
31:that satisfies the
3115:
3083:
3059:
3031:
3011:injective modules.
2946:. You can help by
2889:
2826:
2801:
2764:
2732:
2710:
2685:
2589:
2566:
2518:
2498:
2474:
2454:
2434:
2405:
2385:
2362:
2303:
2240:
2216:
2184:
2151:
2131:
2107:
2087:
2059:
2036:
2013:
1981:algebraic geometry
1865:
1805:
1669:rational functions
1655:is not Noetherian.
1591:algebraic integers
1490:
1446:
1103:
1040:
967:
950:with coefficients
940:
878:linear combination
866:
839:
815:
731:
669:
631:finitely generated
171:finitely generated
157:
96:
35:on left and right
3759:"Noetherian ring"
3744:978-0-521-36764-6
3714:978-0-201-55540-0
3638:978-0-8176-4363-8
3541:978-0-387-19371-7
3526:Bourbaki, Nicolas
3520:978-0-201-40751-8
3451:, Corollary 26.3.
3375:978-0-7923-1394-6
3204:Lam (2001), p. 19
3178:Noetherian scheme
3162:Azumaya's theorem
3154:endomorphism ring
2980:injective modules
2964:
2963:
2628:of finitely many
2609:commutative rings
2592:{\displaystyle G}
2569:{\displaystyle R}
2521:{\displaystyle G}
2501:{\displaystyle R}
2488:. Conversely, if
2477:{\displaystyle G}
2457:{\displaystyle R}
2437:{\displaystyle R}
2408:{\displaystyle R}
2388:{\displaystyle G}
2243:{\displaystyle R}
2219:{\displaystyle R}
2187:{\displaystyle R}
2154:{\displaystyle R}
2134:{\displaystyle G}
2110:{\displaystyle R}
2090:{\displaystyle R}
2062:{\displaystyle R}
2039:{\displaystyle G}
2016:{\displaystyle R}
1512:, are Noetherian.
1464:is a quotient of
1409:rings of integers
1398:Euclidean domains
1190:Noetherian module
1161:ring homomorphism
1122:power series ring
1106:{\displaystyle R}
1043:{\displaystyle R}
842:{\displaystyle n}
779:Given a sequence
754:partially ordered
621:Every left ideal
608:commutative rings
572:Characterizations
569:
568:
3793:
3772:
3747:
3725:
3695:
3657:
3615:
3605:
3580:Formanek, Edward
3575:
3545:
3511:
3464:
3458:
3452:
3446:
3440:
3434:
3428:
3422:
3416:
3410:
3404:
3403:
3353:
3347:
3341:
3335:
3330:
3324:
3318:
3312:
3306:
3300:
3294:
3285:
3279:
3273:
3267:
3261:
3260:
3223:
3217:
3211:
3205:
3202:
3124:
3122:
3121:
3116:
3114:
3113:
3092:
3090:
3089:
3084:
3082:
3081:
3068:
3066:
3065:
3060:
3058:
3057:
3040:
3038:
3037:
3032:
3030:
3029:
2968:Goldie's theorem
2959:
2956:
2938:
2931:
2912:dimension theory
2898:
2896:
2895:
2890:
2882:
2881:
2869:
2868:
2845:ArtinâRees lemma
2835:
2833:
2832:
2827:
2821:
2820:
2819:
2809:
2784:
2783:
2782:
2772:
2741:
2739:
2738:
2733:
2730:
2729:
2728:
2718:
2705:
2704:
2703:
2693:
2615:Commutative case
2598:
2596:
2595:
2590:
2578:Noetherian group
2575:
2573:
2572:
2567:
2541:polycyclic group
2527:
2525:
2524:
2519:
2507:
2505:
2504:
2499:
2486:Noetherian group
2483:
2481:
2480:
2475:
2463:
2461:
2460:
2455:
2443:
2441:
2440:
2435:
2414:
2412:
2411:
2406:
2394:
2392:
2391:
2386:
2371:
2369:
2368:
2363:
2339:
2338:
2312:
2310:
2309:
2304:
2299:
2298:
2249:
2247:
2246:
2241:
2225:
2223:
2222:
2217:
2193:
2191:
2190:
2185:
2160:
2158:
2157:
2152:
2140:
2138:
2137:
2132:
2116:
2114:
2113:
2108:
2096:
2094:
2093:
2088:
2068:
2066:
2065:
2060:
2045:
2043:
2042:
2037:
2022:
2020:
2019:
2014:
1874:
1872:
1871:
1866:
1861:
1857:
1856:
1842:
1828:
1816:
1812:
1810:
1809:
1589:The ring of all
1499:
1497:
1496:
1491:
1486:
1485:
1455:
1453:
1452:
1447:
1445:
1444:
1375:rational numbers
1151:
1138:
1132:
1119:
1112:
1110:
1109:
1104:
1099:
1098:
1080:
1079:
1049:
1047:
1046:
1041:
976:
974:
973:
968:
966:
965:
949:
947:
946:
941:
939:
938:
929:
928:
918:
913:
895:
894:
875:
873:
872:
867:
865:
864:
848:
846:
845:
840:
824:
822:
821:
816:
808:
807:
795:
794:
740:
738:
737:
732:
730:
729:
708:
707:
678:
676:
675:
670:
668:
667:
649:
648:
593:right-Noetherian
561:
554:
547:
336:Commutative ring
265:Rack and quandle
230:
229:
190:polynomial rings
186:ring of integers
166:
164:
163:
158:
147:
146:
128:
127:
111:
105:
103:
102:
97:
89:
88:
76:
75:
63:
62:
45:right-Noetherian
3801:
3800:
3796:
3795:
3794:
3792:
3791:
3790:
3776:
3775:
3757:
3754:
3745:
3715:
3684:
3639:
3603:10.2307/2039890
3572:
3550:Eisenbud, David
3542:
3501:
3473:
3468:
3467:
3459:
3455:
3447:
3443:
3439:, Theorem 25.8.
3435:
3431:
3423:
3419:
3411:
3407:
3376:
3354:
3350:
3342:
3338:
3331:
3327:
3319:
3315:
3307:
3303:
3295:
3288:
3280:
3276:
3268:
3264:
3227:Cohen, Irvin S.
3224:
3220:
3216:, Exercise 1.1.
3212:
3208:
3203:
3196:
3191:
3174:
3109:
3108:
3106:
3103:
3102:
3077:
3076:
3074:
3071:
3070:
3053:
3052:
3050:
3047:
3046:
3025:
3024:
3022:
3019:
3018:
3016:cardinal number
2976:
2960:
2954:
2951:
2944:needs expansion
2929:
2877:
2873:
2864:
2860:
2852:
2849:
2848:
2815:
2811:
2810:
2805:
2778:
2774:
2773:
2768:
2747:
2744:
2743:
2724:
2720:
2719:
2714:
2699:
2695:
2694:
2689:
2677:
2674:
2673:
2617:
2605:
2584:
2581:
2580:
2552:
2549:
2548:
2513:
2510:
2509:
2493:
2490:
2489:
2469:
2466:
2465:
2449:
2446:
2445:
2420:
2417:
2416:
2400:
2397:
2396:
2395:be a group and
2380:
2377:
2376:
2331:
2327:
2319:
2316:
2315:
2294:
2290:
2264:
2261:
2260:
2235:
2232:
2231:
2202:
2199:
2198:
2170:
2167:
2166:
2146:
2143:
2142:
2126:
2123:
2122:
2102:
2099:
2098:
2082:
2079:
2078:
2054:
2051:
2050:
2031:
2028:
2027:
1999:
1996:
1995:
1989:
1927:Eakin's theorem
1852:
1838:
1824:
1804:
1803:
1798:
1792:
1791:
1786:
1776:
1775:
1774:
1771:
1770:
1766:
1758:
1755:
1754:
1661:integral domain
1647:
1640:
1633:
1605:
1585:
1578:
1571:
1564:
1557:
1550:
1543:
1536:
1529:
1481:
1480:
1469:
1466:
1465:
1440:
1439:
1437:
1434:
1433:
1416:coordinate ring
1405:Dedekind domain
1383:complex numbers
1370:
1247:faithfully flat
1143:
1134:
1128:
1114:
1094:
1090:
1075:
1071:
1063:
1060:
1059:
1026:
1023:
1022:
1020:polynomial ring
1011:
961:
957:
955:
952:
951:
934:
930:
924:
920:
914:
903:
890:
886:
884:
881:
880:
860:
856:
854:
851:
850:
849:such that each
834:
831:
830:
825:of elements in
803:
799:
790:
786:
784:
781:
780:
758:maximal element
725:
721:
703:
699:
688:
685:
684:
663:
659:
644:
640:
638:
635:
634:
586:left-Noetherian
574:
565:
536:
535:
534:
505:Non-associative
487:
476:
475:
465:
445:
434:
433:
422:Map of lattices
418:
414:Boolean algebra
409:Heyting algebra
383:
372:
371:
365:
346:Integral domain
310:
299:
298:
292:
246:
136:
132:
123:
119:
117:
114:
113:
107:
84:
80:
71:
67:
58:
54:
52:
49:
48:
41:left-Noetherian
25:Noetherian ring
17:
12:
11:
5:
3799:
3789:
3788:
3774:
3773:
3753:
3752:External links
3750:
3749:
3748:
3743:
3726:
3713:
3696:
3682:
3662:Lam, Tsit Yuen
3658:
3637:
3616:
3596:(2): 181â186.
3576:
3570:
3546:
3540:
3522:
3512:
3499:
3472:
3469:
3466:
3465:
3453:
3441:
3429:
3417:
3405:
3374:
3348:
3336:
3325:
3313:
3301:
3286:
3284:, Theorem 3.6.
3282:Matsumura 1989
3274:
3272:, Theorem 3.5.
3270:Matsumura 1989
3262:
3218:
3206:
3193:
3192:
3190:
3187:
3186:
3185:
3180:
3173:
3170:
3150:
3149:
3126:
3112:
3095:generating set
3080:
3056:
3028:
3012:
3009:indecomposable
3001:
2994:
2975:
2972:
2971:
2970:
2962:
2961:
2941:
2939:
2928:
2925:
2924:
2923:
2908:
2888:
2885:
2880:
2876:
2872:
2867:
2863:
2859:
2856:
2841:
2825:
2818:
2814:
2808:
2804:
2800:
2797:
2794:
2791:
2788:
2781:
2777:
2771:
2767:
2763:
2760:
2757:
2754:
2751:
2727:
2723:
2717:
2713:
2709:
2702:
2698:
2692:
2688:
2684:
2681:
2630:primary ideals
2616:
2613:
2604:
2601:
2588:
2565:
2562:
2559:
2556:
2537:solvable group
2517:
2497:
2473:
2453:
2433:
2430:
2427:
2424:
2404:
2384:
2373:
2372:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2337:
2334:
2330:
2326:
2323:
2313:
2302:
2297:
2293:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2239:
2228:
2227:
2215:
2212:
2209:
2206:
2195:
2183:
2180:
2177:
2174:
2150:
2130:
2121:. For a group
2106:
2086:
2058:
2035:
2012:
2009:
2006:
2003:
1988:
1985:
1977:valuation ring
1917:-module, then
1876:
1875:
1864:
1860:
1855:
1851:
1848:
1845:
1841:
1837:
1834:
1831:
1827:
1823:
1820:
1815:
1808:
1802:
1799:
1797:
1794:
1793:
1790:
1787:
1785:
1782:
1781:
1779:
1773:
1769:
1765:
1762:
1701:
1700:
1657:
1656:
1649:
1645:
1638:
1631:
1619:) = 0 for all
1603:
1594:
1587:
1583:
1576:
1569:
1562:
1555:
1548:
1541:
1534:
1527:
1517:
1516:
1513:
1489:
1484:
1479:
1476:
1473:
1443:
1423:
1420:affine variety
1412:
1401:
1386:
1369:
1366:
1365:
1364:
1329:
1318:
1311:indecomposable
1299:
1284:
1273:
1266:
1231:
1200:
1193:
1179:if and only if
1171:
1164:
1125:
1102:
1097:
1093:
1089:
1086:
1083:
1078:
1074:
1070:
1067:
1039:
1036:
1033:
1030:
1010:
1007:
991:maximal ideals
983:
982:
964:
960:
937:
933:
927:
923:
917:
912:
909:
906:
902:
898:
893:
889:
863:
859:
838:
814:
811:
806:
802:
798:
793:
789:
762:
761:
742:
728:
724:
720:
717:
714:
711:
706:
702:
698:
695:
692:
666:
662:
658:
655:
652:
647:
643:
604:
603:
596:
589:
573:
570:
567:
566:
564:
563:
556:
549:
541:
538:
537:
533:
532:
527:
522:
517:
512:
507:
502:
496:
495:
494:
488:
482:
481:
478:
477:
474:
473:
470:Linear algebra
464:
463:
458:
453:
447:
446:
440:
439:
436:
435:
432:
431:
428:Lattice theory
424:
417:
416:
411:
406:
401:
396:
391:
385:
384:
378:
377:
374:
373:
364:
363:
358:
353:
348:
343:
338:
333:
328:
323:
318:
312:
311:
305:
304:
301:
300:
291:
290:
285:
280:
274:
273:
272:
267:
262:
253:
247:
241:
240:
237:
236:
182:noncommutative
156:
153:
150:
145:
142:
139:
135:
131:
126:
122:
95:
92:
87:
83:
79:
74:
70:
66:
61:
57:
15:
9:
6:
4:
3:
2:
3798:
3787:
3784:
3783:
3781:
3770:
3766:
3765:
3760:
3756:
3755:
3746:
3740:
3736:
3732:
3727:
3724:
3720:
3716:
3710:
3706:
3702:
3698:Chapter X of
3697:
3693:
3689:
3685:
3679:
3675:
3671:
3667:
3663:
3659:
3656:
3652:
3648:
3644:
3640:
3634:
3630:
3626:
3622:
3617:
3613:
3609:
3604:
3599:
3595:
3591:
3590:
3585:
3581:
3577:
3573:
3571:0-387-94268-8
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3537:
3533:
3532:
3527:
3523:
3521:
3517:
3513:
3510:
3506:
3502:
3500:0-387-97845-3
3496:
3492:
3488:
3484:
3480:
3475:
3474:
3463:, Lemma 25.4.
3462:
3457:
3450:
3445:
3438:
3433:
3426:
3421:
3414:
3413:Eisenbud 1995
3409:
3401:
3397:
3393:
3389:
3385:
3381:
3377:
3371:
3367:
3363:
3359:
3352:
3345:
3340:
3334:
3329:
3322:
3317:
3310:
3309:Bourbaki 1989
3305:
3298:
3293:
3291:
3283:
3278:
3271:
3266:
3258:
3254:
3250:
3246:
3242:
3238:
3237:
3232:
3228:
3222:
3215:
3214:Eisenbud 1995
3210:
3201:
3199:
3194:
3184:
3183:Artinian ring
3181:
3179:
3176:
3175:
3169:
3167:
3163:
3159:
3155:
3147:
3143:
3139:
3135:
3131:
3127:
3100:
3096:
3044:
3017:
3013:
3010:
3006:
3002:
2999:
2995:
2992:
2989:
2988:
2987:
2985:
2981:
2969:
2966:
2965:
2958:
2955:December 2019
2949:
2945:
2942:This section
2940:
2937:
2933:
2932:
2921:
2917:
2913:
2909:
2906:
2902:
2886:
2883:
2878:
2874:
2870:
2865:
2861:
2857:
2854:
2846:
2842:
2839:
2816:
2812:
2806:
2802:
2795:
2792:
2789:
2779:
2775:
2769:
2765:
2758:
2752:
2725:
2721:
2715:
2711:
2707:
2700:
2696:
2690:
2686:
2682:
2679:
2671:
2667:
2663:
2659:
2655:
2651:
2647:
2644:and whenever
2643:
2639:
2635:
2631:
2627:
2623:
2619:
2618:
2612:
2610:
2600:
2586:
2579:
2560:
2554:
2546:
2542:
2538:
2535:
2531:
2515:
2495:
2487:
2471:
2451:
2428:
2422:
2402:
2382:
2359:
2353:
2350:
2347:
2335:
2332:
2328:
2321:
2314:
2300:
2295:
2287:
2281:
2272:
2266:
2259:
2258:
2257:
2256:
2253:
2237:
2210:
2204:
2196:
2178:
2172:
2164:
2163:
2162:
2148:
2128:
2120:
2104:
2084:
2076:
2072:
2056:
2049:
2033:
2026:
2007:
2001:
1994:
1991:Consider the
1984:
1982:
1978:
1973:
1971:
1967:
1962:
1960:
1957:-module, but
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1908:
1904:
1899:
1897:
1893:
1889:
1885:
1881:
1862:
1858:
1849:
1846:
1843:
1835:
1832:
1829:
1821:
1818:
1813:
1806:
1800:
1795:
1788:
1783:
1777:
1767:
1763:
1760:
1753:
1752:
1751:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1721:
1717:
1714:
1710:
1706:
1698:
1694:
1690:
1686:
1683:over a field
1682:
1678:
1674:
1671:generated by
1670:
1666:
1665:
1664:
1662:
1654:
1650:
1644:
1637:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1599:
1595:
1592:
1588:
1582:
1575:
1568:
1561:
1554:
1547:
1540:
1533:
1526:
1522:
1521:
1520:
1514:
1511:
1507:
1503:
1474:
1471:
1463:
1459:
1432:
1428:
1424:
1421:
1417:
1413:
1410:
1406:
1402:
1399:
1395:
1391:
1387:
1384:
1380:
1376:
1372:
1371:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1327:
1323:
1319:
1316:
1312:
1308:
1304:
1300:
1297:
1293:
1289:
1285:
1282:
1281:Artinian ring
1278:
1274:
1271:
1267:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1229:
1225:
1221:
1217:
1213:
1209:
1205:
1201:
1198:
1194:
1191:
1187:
1185:
1180:
1176:
1172:
1169:
1165:
1162:
1159:
1155:
1150:
1146:
1142:
1141:quotient ring
1137:
1131:
1126:
1123:
1117:
1095:
1091:
1087:
1084:
1081:
1076:
1072:
1065:
1057:
1053:
1034:
1028:
1021:
1017:
1013:
1012:
1006:
1004:
1000:
996:
992:
988:
980:
962:
958:
935:
931:
925:
921:
915:
910:
907:
904:
900:
896:
891:
887:
879:
861:
857:
836:
828:
812:
809:
804:
800:
796:
791:
787:
778:
777:
776:
774:
770:
765:
759:
755:
751:
747:
743:
726:
722:
718:
715:
712:
709:
704:
700:
696:
693:
690:
682:
664:
660:
656:
653:
650:
645:
641:
632:
628:
624:
620:
619:
618:
616:
611:
609:
601:
597:
594:
590:
587:
583:
582:
581:
579:
562:
557:
555:
550:
548:
543:
542:
540:
539:
531:
528:
526:
523:
521:
518:
516:
513:
511:
508:
506:
503:
501:
498:
497:
493:
490:
489:
485:
480:
479:
472:
471:
467:
466:
462:
459:
457:
454:
452:
449:
448:
443:
438:
437:
430:
429:
425:
423:
420:
419:
415:
412:
410:
407:
405:
402:
400:
397:
395:
392:
390:
387:
386:
381:
376:
375:
370:
369:
362:
359:
357:
356:Division ring
354:
352:
349:
347:
344:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
317:
314:
313:
308:
303:
302:
297:
296:
289:
286:
284:
281:
279:
278:Abelian group
276:
275:
271:
268:
266:
263:
261:
257:
254:
252:
249:
248:
244:
239:
238:
235:
232:
231:
228:
226:
222:
218:
217:David Hilbert
214:
209:
207:
203:
199:
198:number fields
195:
191:
187:
183:
179:
174:
172:
167:
154:
151:
148:
143:
140:
137:
133:
129:
124:
120:
110:
93:
90:
85:
81:
77:
72:
68:
64:
59:
55:
46:
42:
38:
34:
30:
26:
22:
3762:
3730:
3704:
3665:
3620:
3593:
3587:
3553:
3530:
3478:
3456:
3444:
3432:
3420:
3408:
3357:
3351:
3339:
3328:
3316:
3304:
3277:
3265:
3243:(1): 27â42.
3240:
3234:
3221:
3209:
3151:
3145:
3137:
3133:
3129:
3042:
3004:
2997:
2990:
2983:
2977:
2952:
2948:adding to it
2943:
2669:
2665:
2661:
2657:
2653:
2649:
2645:
2637:
2626:intersection
2606:
2603:Key theorems
2545:finite group
2374:
2255:homomorphism
2229:
1990:
1974:
1963:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1930:
1922:
1918:
1914:
1910:
1906:
1902:
1900:
1895:
1891:
1887:
1883:
1879:
1877:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1712:
1704:
1702:
1696:
1692:
1688:
1684:
1680:
1676:
1672:
1667:The ring of
1658:
1651:The ring of
1642:
1635:
1628:
1624:
1620:
1616:
1612:
1608:
1601:
1596:The ring of
1580:
1573:
1566:
1559:
1552:
1545:
1538:
1531:
1524:
1518:
1506:Weyl algebra
1461:
1426:
1379:real numbers
1356:
1340:
1332:
1324:. Also, the
1270:localization
1262:
1259:pure subring
1254:
1250:
1242:
1238:
1234:
1227:
1223:
1219:
1215:
1207:
1206:) If a ring
1204:EakinâNagata
1183:
1174:
1148:
1144:
1135:
1129:
1115:
1015:
984:
978:
876:is a finite
826:
768:
766:
763:
749:
680:
626:
622:
614:
612:
605:
599:
592:
585:
575:
530:Hopf algebra
468:
461:Vector space
426:
366:
295:Group theory
293:
258: /
213:Emmy Noether
210:
175:
168:
108:
44:
40:
24:
18:
3786:Ring theory
3701:Lang, Serge
3346:, Theorem 3
3099:cardinality
2415:a ring. If
2119:commutative
1502:PBW theorem
1431:Lie algebra
1339:(in short,
987:prime ideal
515:Lie algebra
500:Associative
404:Total order
394:Semilattice
368:Ring theory
178:commutative
112:such that:
21:mathematics
3723:0848.13001
3683:0387951830
3655:1292.00026
3471:References
3400:0732.20019
2534:Noetherian
2069:. It is a
1993:group ring
1716:isomorphic
1611:such that
1303:direct sum
1296:Ore domain
1294:is a left
1241:such that
1218:such that
1158:surjective
1009:Properties
995:local ring
683:such that
600:Noetherian
598:A ring is
591:A ring is
584:A ring is
3769:EMS Press
3384:0169-6378
3257:0012-7094
3160:and thus
2887:⋯
2884:⊇
2871:⊇
2858:⊇
2796:∩
2793:⋯
2790:∩
2708:⋯
2652:, either
2530:extension
2351:∈
2345:∀
2333:−
2325:↦
2279:→
2197:The ring
2165:The ring
2073:, and an
1898:-module.
1850:∈
1847:γ
1836:∈
1833:β
1822:∈
1801:γ
1789:β
1475:
1307:injective
1085:…
1056:induction
999:principal
901:∑
813:…
746:non-empty
713:⋯
654:…
525:Bialgebra
331:Near-ring
288:Lie group
256:Semigroup
152:⋯
94:⋯
91:⊆
78:⊆
65:⊆
3780:Category
3703:(1993),
3664:(2001).
3552:(1995).
3528:(1989).
3229:(1950).
3172:See also
3140:-module
3132:-module
3101:at most
2634:radicals
2539:(i.e. a
1890:= 0 and
1709:subgroup
1679: /
1368:Examples
1288:coherent
1261:), then
1197:faithful
361:Lie ring
326:Semiring
204:and the
3771:, 2001
3705:Algebra
3692:1838439
3647:2357361
3612:2039890
3509:1245487
3392:1191619
2632:(whose
2547:, then
2543:) by a
2046:over a
1949:), and
1407:(e.g.,
1355:, then
1226:, then
1212:subring
1186:-module
1173:A ring
1156:of any
773:Hilbert
492:Algebra
484:Algebra
389:Lattice
380:Lattice
3741:
3721:
3711:
3690:
3680:
3653:
3645:
3635:
3610:
3568:
3538:
3518:
3507:
3497:
3398:
3390:
3382:
3372:
3255:
3142:embeds
2642:proper
2528:is an
1941:= Hom(
1909:, and
1722:, let
1418:of an
1381:, and
1317:below.
1292:domain
1268:Every
1120:, the
744:Every
520:Graded
451:Module
442:Module
341:Domain
260:Monoid
192:, and
37:ideals
3608:JSTOR
3189:Notes
3158:local
2901:prove
2532:of a
2484:is a
2077:over
2025:group
2023:of a
1730:from
1707:is a
1359:is a
1353:units
1349:up to
1343:is a
1257:as a
1249:over
1210:is a
1188:is a
1154:image
1054:. By
486:-like
444:-like
382:-like
351:Field
309:-like
283:Magma
251:Group
245:-like
243:Group
27:is a
3739:ISBN
3709:ISBN
3678:ISBN
3633:ISBN
3566:ISBN
3536:ISBN
3516:ISBN
3495:ISBN
3380:ISSN
3370:ISBN
3253:ISSN
3152:The
2910:The
2375:Let
2071:ring
2048:ring
1742:) â
1675:and
1565:), (
1551:), (
1414:The
1396:and
1388:Any
606:For
576:For
316:Ring
307:Ring
180:and
29:ring
23:, a
3719:Zbl
3670:doi
3651:Zbl
3625:doi
3598:doi
3558:doi
3487:doi
3396:Zbl
3362:doi
3245:doi
3168:).
3097:of
2950:.
2660:or
2117:is
2097:if
1901:If
1750:as
1718:to
1711:of
1472:Sym
1460:of
1305:of
1245:is
1127:If
1014:If
1005:.)
977:in
679:in
629:is
625:in
321:Rng
208:).
196:in
43:or
19:In
3782::
3767:,
3761:,
3737:,
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3688:MR
3686:.
3676:.
3649:,
3643:MR
3641:,
3631:,
3606:.
3594:46
3592:.
3586:.
3564:.
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3493:,
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3378:.
3368:.
3289:^
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3233:.
3197:^
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2648:â
2646:xy
2296:op
1975:A
1964:A
1945:,
1882:â
1641:,
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1623:â„
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227:.
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3574:.
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3364::
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3146:H
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2907:.
2879:3
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2807:r
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2799:(
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2762:(
2759:=
2756:)
2753:f
2750:(
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2270:[
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2208:[
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2176:[
2173:R
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2034:G
2011:]
2008:G
2005:[
2002:R
1959:R
1955:R
1951:S
1947:Q
1943:Q
1939:S
1935:R
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1911:S
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1202:(
1192:.
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149:=
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