872:
614:
1165:. We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an
249:, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal. Before about 1960 many authors required that a local ring be (left and right)
192:. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring
2150:
1255:
With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs
2324:
2345:(which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of
1722:
328:
459:
686:
2625:
2539:
464:
1827:
2577:
401:
1372:
1120:
1094:
1024:
2176:
2208:
1885:
1854:
2234:
1783:
1050:
999:
1070:
973:
953:
933:
688:, is local. Its unique maximal ideal consists of all elements that are not invertible. In other words, it consists of all elements with constant term zero.
184:
If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).
2030:
2454:
17:
2239:
2787:
The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals.
1382:
188:
If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's
2697:
789:
denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is the integers
2834:
729:) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of
46:
that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on
2897:
1667:
2812:
291:
2349:. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.
406:
1177:
of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.
619:
609:{\textstyle (\sum _{i=0}^{\infty }a_{i}x^{i})(\sum _{i=0}^{\infty }b_{i}x^{i})=\sum _{i=0}^{\infty }c_{i}x^{i}}
2586:
2500:
1788:
2544:
2019:. These are precisely the ring homomorphisms that are continuous with respect to the given topologies on
2482:
1911:
1562:
369:
1355:
1103:
1077:
1007:
2850:
1915:
1279:
1270:
Exactly the same arguments work for the ring of germs of continuous real-valued functions on any
1180:
To see that this ring of germs is local, we need to characterize its invertible elements. A germ
883:
356:
55:
2155:
1903:
1158:
815:
790:
360:
82:
2865:
1539:
1523:
2892:
2181:
1863:
1832:
1744:
1527:
1295:
1291:
1166:
703:
75:
1290:
at a given point. All these rings are therefore local. These examples help to explain why
8:
2470:
2213:
1762:
1748:
1558:
1307:
1154:
1029:
978:
738:
692:
278:
67:
59:
47:
2722:
2675:
2474:
1547:
1466:
1174:
1148:
1055:
958:
938:
909:
117:
43:
1755:
be the ring of germs of infinitely differentiable functions at 0 in the real line and
2830:
2808:
2679:
2462:
1989:
1631:
1519:
1390:
1287:
1283:
1271:
1170:
2712:
2667:
2458:
2335:
1623:
797:
734:
246:
189:
71:
2469:, though the case where the module is finitely-generated is a simple corollary to
695:
over a local ring is local; the maximal ideal consists of those power series with
2855:
2822:
1740:
1658:
1635:
1345:
904:
782:
261:
250:
201:
2860:
2750:
2655:
1313:
1275:
152:
93:
2671:
1510:
is a concept without a simple definition. It is replaced by using valuations.
2886:
2693:
1941:
1907:
719:
696:
135:
1914:, again a local ring. Complete Noetherian local rings are classified by the
1530:
over some other rings. Specifically, if the endomorphism ring of the module
2774:
707:
2631:, the conclusion is that the only rings Morita equivalent to a local ring
830:
in this localization; that is, the maximal ideal consists of all elements
348:
More generally, a nonzero ring in which every element is either a unit or
2466:
2406:
2145:{\displaystyle \mathbb {C} /(x^{3})\to \mathbb {C} /(x^{3},x^{2}y,y^{4})}
804:
86:
35:
31:
1416:
has dimension 2 or more there is a difficulty that is seen this way: if
871:
81:
In practice, a commutative local ring often arises as the result of the
2726:
2417:
1583:
1349:
786:
761:
282:
139:
1153:
To motivate the name "local" for these rings, we consider real-valued
285:) are local rings, since {0} is the only maximal ideal in these rings.
1162:
757:
349:
2877:
2717:
1751:, and, as such, the "Noetherian" assumption is crucial. Indeed, let
1312:
Local rings play a major role in valuation theory. By definition, a
2800:
51:
1344:. Any such subring will be a local ring. For example, the ring of
2807:. Graduate Texts in Mathematics (2nd ed.). Springer-Verlag.
1576:
197:
105:
2698:"Foundations of a General Theory of Birational Correspondences"
1198:, then by continuity there is an open interval around 0 where
2319:{\displaystyle \mathbb {C} /(x^{3})\to \mathbb {C} /(x^{2})}
1550:
and is indecomposable, then its endomorphism ring is local.
1294:, the generalizations of varieties, are defined as special
127:
if it has any one of the following equivalent properties:
341:). The unique maximal ideal consists of all multiples of
253:, and (possibly non-Noetherian) local rings were called
2579:. Since every ring Morita equivalent to the local ring
2236:. Another example of a local ring morphism is given by
1352:
denominator (mentioned above) is a valuation ring in
826:
is local; the maximal ideal is the ideal generated by
622:
467:
409:
2589:
2547:
2503:
2242:
2216:
2184:
2158:
2033:
1866:
1835:
1791:
1765:
1670:
1358:
1106:
1080:
1058:
1032:
1010:
981:
961:
941:
912:
372:
294:
111:
1717:{\displaystyle \bigcap _{i=1}^{\infty }m^{i}=\{0\}}
257:. In this article this requirement is not imposed.
2619:
2571:
2533:
2473:. This has an interesting consequence in terms of
2318:
2228:
2202:
2170:
2144:
1879:
1848:
1821:
1777:
1716:
1366:
1114:
1088:
1064:
1044:
1018:
993:
967:
947:
927:
680:
608:
453:
395:
322:
1890:As for any topological ring, one can ask whether
2884:
1945:of the local ring or residue field of the point
778:Nonzero quotient rings of local rings are local.
1518:Non-commutative local rings arise naturally as
196:is local if and only if there do not exist two
27:(Mathematical) ring with a unique maximal ideal
2711:(3). American Mathematical Society: 490–542 .
2658:(1938). "Dimensionstheorie in Stellenringen".
323:{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
1592:
92:The concept of local rings was introduced by
1925:is the local ring of a scheme at some point
1711:
1705:
1626:in a natural way if one takes the powers of
710:over any field is local. More generally, if
454:{\textstyle \sum _{i=0}^{\infty }a_{i}x^{i}}
2541:is isomorphic to the full ring of matrices
2635:are (isomorphic to) the matrix rings over
2027:. For example, consider the ring morphism
1282:at a given point, or the ring of germs of
1274:at a given point, or the ring of germs of
1202:is non-zero, and we can form the function
1026:is not local since it has a maximal ideal
681:{\textstyle c_{n}=\sum _{i+j=n}a_{i}b_{j}}
2716:
2282:
2244:
2073:
2035:
1360:
1225:gives rise to a germ, and the product of
1108:
1082:
1012:
374:
316:
296:
2821:
1473:. Considering a simple example, such as
1404:we could try to define a valuation ring
1385:, we may look for local rings in it. If
2692:
1921:In algebraic geometry, especially when
1001:are non-units, but their sum is a unit.
14:
2885:
2805:A first course in noncommutative rings
2686:
2497:, and hence the ring of endomorphisms
1743:. The theorem is a consequence of the
699:in the maximal ideal of the base ring.
355:An important class of local rings are
2829:. Vol. 2 (2nd ed.). Dover.
2748:
2654:
2620:{\displaystyle \mathrm {End} _{R}(P)}
2534:{\displaystyle \mathrm {End} _{R}(P)}
1389:were indeed the function field of an
1324:such that for every non-zero element
403:, whose elements are infinite series
2648:
1822:{\displaystyle e^{-{1 \over x^{2}}}}
866:
2799:
2572:{\displaystyle \mathrm {M} _{n}(R)}
2442:is again local, with maximal ideal
1910:); if it is not, one considers its
1597:
1301:
461:where multiplications are given by
24:
2598:
2595:
2592:
2550:
2512:
2509:
2506:
1687:
1513:
1233:is invertible, then there is some
1130:are distinct prime numbers. Both (
581:
534:
487:
426:
204:) (left) ideals, where two ideals
25:
2909:
2878:The philosophy behind local rings
2871:
2493:is isomorphic to the free module
1860:, since that function divided by
151:1 ≠ 0 and the sum of any two non-
148:has a unique maximal right ideal.
112:Definition and first consequences
2360:, the following are equivalent:
1142:
870:
737:to invert all other polynomials
718:is a positive integer, then the
2329:
1229:is equal to 1. (Conversely, if
1221:on this interval. The function
862:
2781:
2767:
2742:
2739:Lam (2001), p. 295, Thm. 19.1.
2733:
2614:
2608:
2566:
2560:
2528:
2522:
2313:
2300:
2292:
2286:
2278:
2275:
2262:
2254:
2248:
2223:
2217:
2197:
2185:
2162:
2139:
2097:
2089:
2077:
2069:
2066:
2053:
2045:
2039:
1772:
1766:
1039:
1033:
922:
916:
691:More generally, every ring of
559:
515:
512:
468:
390:
387:
381:
378:
13:
1:
2793:
1614:for a commutative local ring
1381:, which may or may not be a
1184:is invertible if and only if
748:is a field, then elements of
396:{\displaystyle \mathbb {C} ]}
1729:Krull's intersection theorem
1622:. Every such ring becomes a
1542:; conversely, if the module
1367:{\displaystyle \mathbb {Q} }
1115:{\displaystyle \mathbb {Z} }
1089:{\displaystyle \mathbb {Z} }
1019:{\displaystyle \mathbb {Z} }
7:
2844:
271:
10:
2914:
2898:Localization (mathematics)
2430:is any two-sided ideal in
2171:{\displaystyle x\mapsto x}
1785:. Then a nonzero function
1593:Some facts and definitions
1424:are rational functions on
1408:of functions "defined at"
1305:
1146:
1138:) are maximal ideals here.
796:More generally, given any
18:Krull intersection theorem
2672:10.1515/crll.1938.179.204
1173:are what are called the "
764:. (The dual numbers over
58:examined at a particular
2642:
1976:are local rings, then a
1488:approached along a line
357:discrete valuation rings
260:A local ring that is an
2851:Discrete valuation ring
2434:, then the factor ring
1978:local ring homomorphism
1916:Cohen structure theorem
1731:), and it follows that
1280:differentiable manifold
768:correspond to the case
361:principal ideal domains
96:in 1938 under the name
56:algebraic number fields
34:, more specifically in
2705:Trans. Amer. Math. Soc
2621:
2573:
2535:
2320:
2230:
2204:
2172:
2146:
1881:
1850:
1823:
1779:
1718:
1691:
1396:, then for each point
1368:
1116:
1090:
1066:
1046:
1020:
995:
969:
949:
929:
733:, since one can use a
682:
610:
585:
538:
491:
455:
430:
397:
324:
83:localization of a ring
74:local rings and their
2866:Gorenstein local ring
2755:mathworld.wolfram.com
2660:J. Reine Angew. Math.
2622:
2574:
2536:
2465:over a local ring is
2321:
2231:
2205:
2203:{\displaystyle (x,y)}
2173:
2147:
1882:
1880:{\displaystyle x^{n}}
1851:
1849:{\displaystyle m^{n}}
1824:
1780:
1759:be the maximal ideal
1719:
1671:
1369:
1296:locally ringed spaces
1117:
1091:
1067:
1047:
1021:
1004:The ring of integers
996:
970:
950:
930:
683:
611:
565:
518:
471:
456:
410:
398:
325:
2587:
2545:
2501:
2240:
2214:
2182:
2156:
2031:
1864:
1833:
1789:
1763:
1739:-adic topology is a
1668:
1356:
1167:equivalence relation
1155:continuous functions
1104:
1078:
1056:
1030:
1008:
979:
959:
955:is not local, since
939:
910:
714:is a local ring and
620:
465:
407:
370:
363:that are not fields.
292:
100:. The English term
2749:Weisstein, Eric W.
2405:is local, then the
2373:has a right inverse
2229:{\displaystyle (x)}
1778:{\displaystyle (x)}
1618:with maximal ideal
1308:Valuation (algebra)
1171:equivalence classes
1045:{\displaystyle (p)}
994:{\displaystyle 1-x}
905:ring of polynomials
693:formal power series
68:commutative algebra
48:algebraic varieties
2617:
2569:
2531:
2483:finitely generated
2475:Morita equivalence
2367:has a left inverse
2356:of the local ring
2316:
2226:
2200:
2178:. The preimage of
2168:
2142:
2005:with the property
1877:
1846:
1819:
1775:
1714:
1634:of 0. This is the
1526:decompositions of
1520:endomorphism rings
1503:one sees that the
1467:indeterminate form
1364:
1332:, at least one of
1284:rational functions
1149:Germ (mathematics)
1112:
1086:
1062:
1042:
1016:
991:
965:
945:
925:
882:. You can help by
678:
657:
606:
451:
393:
359:, which are local
320:
166:is any element of
2836:978-0-486-47187-7
2463:projective module
1990:ring homomorphism
1887:is still smooth.
1815:
1661:local ring, then
1657:is a commutative
1632:neighborhood base
1412:. In cases where
1391:algebraic variety
1288:algebraic variety
1278:functions on any
1272:topological space
1191:. The reason: if
1065:{\displaystyle p}
968:{\displaystyle x}
948:{\displaystyle K}
928:{\displaystyle K}
900:
899:
636:
330:is a local ring (
255:quasi-local rings
247:commutative rings
66:is the branch of
16:(Redirected from
2905:
2840:
2823:Jacobson, Nathan
2818:
2788:
2785:
2779:
2778:
2771:
2765:
2764:
2762:
2761:
2746:
2740:
2737:
2731:
2730:
2720:
2702:
2690:
2684:
2683:
2652:
2626:
2624:
2623:
2618:
2607:
2606:
2601:
2578:
2576:
2575:
2570:
2559:
2558:
2553:
2540:
2538:
2537:
2532:
2521:
2520:
2515:
2471:Nakayama's lemma
2459:Irving Kaplansky
2429:
2404:
2341:of a local ring
2336:Jacobson radical
2325:
2323:
2322:
2317:
2312:
2311:
2299:
2285:
2274:
2273:
2261:
2247:
2235:
2233:
2232:
2227:
2209:
2207:
2206:
2201:
2177:
2175:
2174:
2169:
2151:
2149:
2148:
2143:
2138:
2137:
2122:
2121:
2109:
2108:
2096:
2076:
2065:
2064:
2052:
2038:
2018:
2004:
1975:
1963:
1938:
1901:
1886:
1884:
1883:
1878:
1876:
1875:
1855:
1853:
1852:
1847:
1845:
1844:
1828:
1826:
1825:
1820:
1818:
1817:
1816:
1814:
1813:
1801:
1784:
1782:
1781:
1776:
1749:Nakayama's lemma
1745:Artin–Rees lemma
1723:
1721:
1720:
1715:
1701:
1700:
1690:
1685:
1656:
1624:topological ring
1613:
1598:Commutative case
1570:
1522:in the study of
1373:
1371:
1370:
1365:
1363:
1346:rational numbers
1302:Valuation theory
1266:
1251:
1220:
1197:
1190:
1161:around 0 of the
1157:defined on some
1121:
1119:
1118:
1113:
1111:
1095:
1093:
1092:
1087:
1085:
1071:
1069:
1068:
1063:
1052:for every prime
1051:
1049:
1048:
1043:
1025:
1023:
1022:
1017:
1015:
1000:
998:
997:
992:
974:
972:
971:
966:
954:
952:
951:
946:
934:
932:
931:
926:
895:
892:
874:
867:
798:commutative ring
783:rational numbers
774:
735:geometric series
687:
685:
684:
679:
677:
676:
667:
666:
656:
632:
631:
615:
613:
612:
607:
605:
604:
595:
594:
584:
579:
558:
557:
548:
547:
537:
532:
511:
510:
501:
500:
490:
485:
460:
458:
457:
452:
450:
449:
440:
439:
429:
424:
402:
400:
399:
394:
377:
352:is a local ring.
344:
340:
333:
329:
327:
326:
321:
319:
314:
313:
304:
299:
241:
190:Jacobson radical
180:
21:
2913:
2912:
2908:
2907:
2906:
2904:
2903:
2902:
2883:
2882:
2874:
2856:Semi-local ring
2847:
2837:
2815:
2796:
2791:
2786:
2782:
2773:
2772:
2768:
2759:
2757:
2747:
2743:
2738:
2734:
2718:10.2307/1990215
2700:
2691:
2687:
2656:Krull, Wolfgang
2653:
2649:
2645:
2602:
2591:
2590:
2588:
2585:
2584:
2583:is of the form
2554:
2549:
2548:
2546:
2543:
2542:
2516:
2505:
2504:
2502:
2499:
2498:
2421:
2394:
2352:For an element
2332:
2307:
2303:
2295:
2281:
2269:
2265:
2257:
2243:
2241:
2238:
2237:
2215:
2212:
2211:
2183:
2180:
2179:
2157:
2154:
2153:
2133:
2129:
2117:
2113:
2104:
2100:
2092:
2072:
2060:
2056:
2048:
2034:
2032:
2029:
2028:
2006:
1992:
1965:
1953:
1930:
1891:
1871:
1867:
1865:
1862:
1861:
1840:
1836:
1834:
1831:
1830:
1809:
1805:
1800:
1796:
1792:
1790:
1787:
1786:
1764:
1761:
1760:
1741:Hausdorff space
1696:
1692:
1686:
1675:
1669:
1666:
1665:
1646:
1603:
1600:
1595:
1565:
1534:is local, then
1516:
1514:Non-commutative
1359:
1357:
1354:
1353:
1310:
1304:
1261:
1246:
1245:(0) = 1, hence
1203:
1192:
1185:
1151:
1145:
1107:
1105:
1102:
1101:
1081:
1079:
1076:
1075:
1057:
1054:
1053:
1031:
1028:
1027:
1011:
1009:
1006:
1005:
980:
977:
976:
960:
957:
956:
940:
937:
936:
911:
908:
907:
896:
890:
887:
880:needs expansion
865:
769:
702:Similarly, the
672:
668:
662:
658:
640:
627:
623:
621:
618:
617:
600:
596:
590:
586:
580:
569:
553:
549:
543:
539:
533:
522:
506:
502:
496:
492:
486:
475:
466:
463:
462:
445:
441:
435:
431:
425:
414:
408:
405:
404:
373:
371:
368:
367:
342:
335:
331:
315:
309:
305:
300:
295:
293:
290:
289:
274:
262:integral domain
245:In the case of
240:
233:
223:
217:
210:
175:
114:
28:
23:
22:
15:
12:
11:
5:
2911:
2901:
2900:
2895:
2881:
2880:
2873:
2872:External links
2870:
2869:
2868:
2863:
2861:Valuation ring
2858:
2853:
2846:
2843:
2842:
2841:
2835:
2819:
2813:
2795:
2792:
2790:
2789:
2780:
2766:
2741:
2732:
2694:Zariski, Oscar
2685:
2646:
2644:
2641:
2616:
2613:
2610:
2605:
2600:
2597:
2594:
2568:
2565:
2562:
2557:
2552:
2530:
2527:
2524:
2519:
2514:
2511:
2508:
2461:says that any
2391:
2390:
2380:
2374:
2368:
2331:
2328:
2315:
2310:
2306:
2302:
2298:
2294:
2291:
2288:
2284:
2280:
2277:
2272:
2268:
2264:
2260:
2256:
2253:
2250:
2246:
2225:
2222:
2219:
2199:
2196:
2193:
2190:
2187:
2167:
2164:
2161:
2141:
2136:
2132:
2128:
2125:
2120:
2116:
2112:
2107:
2103:
2099:
2095:
2091:
2088:
2085:
2082:
2079:
2075:
2071:
2068:
2063:
2059:
2055:
2051:
2047:
2044:
2041:
2037:
1939:is called the
1874:
1870:
1843:
1839:
1812:
1808:
1804:
1799:
1795:
1774:
1771:
1768:
1747:together with
1725:
1724:
1713:
1710:
1707:
1704:
1699:
1695:
1689:
1684:
1681:
1678:
1674:
1639:-adic topology
1602:We also write
1599:
1596:
1594:
1591:
1563:characteristic
1540:indecomposable
1515:
1512:
1501:
1500:
1486:
1485:
1463:
1462:
1449:
1448:
1383:function field
1377:Given a field
1362:
1314:valuation ring
1306:Main article:
1303:
1300:
1276:differentiable
1147:Main article:
1144:
1141:
1140:
1139:
1110:
1084:
1073:
1061:
1041:
1038:
1035:
1014:
1002:
990:
987:
984:
964:
944:
924:
921:
918:
915:
898:
897:
877:
875:
864:
861:
860:
859:
794:
779:
776:
700:
689:
675:
671:
665:
661:
655:
652:
649:
646:
643:
639:
635:
630:
626:
603:
599:
593:
589:
583:
578:
575:
572:
568:
564:
561:
556:
552:
546:
542:
536:
531:
528:
525:
521:
517:
514:
509:
505:
499:
495:
489:
484:
481:
478:
474:
470:
448:
444:
438:
434:
428:
423:
420:
417:
413:
392:
389:
386:
383:
380:
376:
364:
353:
346:
318:
312:
308:
303:
298:
286:
273:
270:
238:
231:
215:
208:
186:
185:
182:
160:
159:is a non-unit.
149:
143:
113:
110:
94:Wolfgang Krull
26:
9:
6:
4:
3:
2:
2910:
2899:
2896:
2894:
2891:
2890:
2888:
2879:
2876:
2875:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2848:
2838:
2832:
2828:
2827:Basic algebra
2824:
2820:
2816:
2814:0-387-95183-0
2810:
2806:
2802:
2798:
2797:
2784:
2776:
2770:
2756:
2752:
2745:
2736:
2728:
2724:
2719:
2714:
2710:
2706:
2699:
2695:
2689:
2681:
2677:
2673:
2669:
2665:
2662:(in German).
2661:
2657:
2651:
2647:
2640:
2638:
2634:
2630:
2611:
2603:
2582:
2563:
2555:
2525:
2517:
2496:
2492:
2489:module, then
2488:
2484:
2480:
2477:. Namely, if
2476:
2472:
2468:
2464:
2460:
2456:
2451:
2449:
2445:
2441:
2437:
2433:
2428:
2424:
2419:
2415:
2411:
2408:
2402:
2398:
2388:
2384:
2381:
2379:is invertible
2378:
2375:
2372:
2369:
2366:
2363:
2362:
2361:
2359:
2355:
2350:
2348:
2344:
2340:
2337:
2327:
2308:
2304:
2296:
2289:
2270:
2266:
2258:
2251:
2220:
2194:
2191:
2188:
2165:
2159:
2134:
2130:
2126:
2123:
2118:
2114:
2110:
2105:
2101:
2093:
2086:
2083:
2080:
2061:
2057:
2049:
2042:
2026:
2022:
2017:
2013:
2009:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1973:
1969:
1961:
1957:
1950:
1948:
1944:
1943:
1942:residue field
1937:
1933:
1928:
1924:
1919:
1917:
1913:
1909:
1908:uniform space
1905:
1899:
1895:
1888:
1872:
1868:
1859:
1841:
1837:
1810:
1806:
1802:
1797:
1793:
1769:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1708:
1702:
1697:
1693:
1682:
1679:
1676:
1672:
1664:
1663:
1662:
1660:
1654:
1650:
1644:
1640:
1638:
1633:
1629:
1625:
1621:
1617:
1611:
1607:
1590:
1588:
1585:
1584:group algebra
1581:
1579:
1574:
1568:
1564:
1560:
1556:
1551:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1511:
1509:
1506:
1498:
1494:
1491:
1490:
1489:
1483:
1479:
1476:
1475:
1474:
1472:
1468:
1461:
1457:
1454:
1453:
1452:
1451:the function
1446:
1442:
1438:
1434:
1431:
1430:
1429:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1392:
1388:
1384:
1380:
1375:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1320:is a subring
1319:
1315:
1309:
1299:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1268:
1264:
1259:
1253:
1249:
1244:
1240:
1236:
1232:
1228:
1224:
1218:
1214:
1210:
1206:
1201:
1195:
1188:
1183:
1178:
1176:
1172:
1168:
1164:
1160:
1159:open interval
1156:
1150:
1143:Ring of germs
1137:
1133:
1129:
1125:
1099:
1074:
1059:
1036:
1003:
988:
985:
982:
962:
942:
935:over a field
919:
913:
906:
902:
901:
894:
885:
881:
878:This section
876:
873:
869:
868:
857:
853:
849:
845:
841:
837:
833:
829:
825:
821:
817:
813:
809:
806:
802:
799:
795:
792:
788:
784:
780:
777:
772:
767:
763:
759:
756:) are either
755:
751:
747:
743:
740:
736:
732:
728:
724:
721:
720:quotient ring
717:
713:
709:
705:
701:
698:
697:constant term
694:
690:
673:
669:
663:
659:
653:
650:
647:
644:
641:
637:
633:
628:
624:
601:
597:
591:
587:
576:
573:
570:
566:
562:
554:
550:
544:
540:
529:
526:
523:
519:
507:
503:
497:
493:
482:
479:
476:
472:
446:
442:
436:
432:
421:
418:
415:
411:
384:
365:
362:
358:
354:
351:
347:
338:
310:
306:
301:
287:
284:
280:
276:
275:
269:
267:
263:
258:
256:
252:
248:
243:
237:
230:
226:
221:
214:
207:
203:
199:
195:
191:
183:
179:
173:
169:
165:
162:1 ≠ 0 and if
161:
158:
154:
150:
147:
144:
141:
137:
134:has a unique
133:
130:
129:
128:
126:
122:
119:
109:
107:
103:
99:
95:
90:
88:
84:
79:
77:
73:
70:that studies
69:
65:
64:Local algebra
61:
57:
53:
49:
45:
41:
37:
33:
19:
2826:
2804:
2783:
2769:
2758:. Retrieved
2754:
2751:"Local Ring"
2744:
2735:
2708:
2704:
2696:(May 1943).
2688:
2666:(179): 204.
2663:
2659:
2650:
2636:
2632:
2628:
2580:
2494:
2490:
2486:
2478:
2455:deep theorem
2452:
2447:
2443:
2439:
2435:
2431:
2426:
2422:
2413:
2409:
2400:
2396:
2392:
2386:
2382:
2376:
2370:
2364:
2357:
2353:
2351:
2346:
2342:
2338:
2333:
2330:General case
2024:
2020:
2015:
2011:
2007:
2001:
1997:
1993:
1985:
1981:
1977:
1971:
1967:
1959:
1955:
1951:
1946:
1940:
1935:
1931:
1926:
1922:
1920:
1897:
1893:
1889:
1857:
1756:
1752:
1736:
1732:
1728:
1726:
1652:
1648:
1642:
1636:
1627:
1619:
1615:
1609:
1605:
1601:
1586:
1577:
1575:is a finite
1572:
1566:
1554:
1552:
1543:
1535:
1531:
1517:
1507:
1504:
1502:
1496:
1492:
1487:
1481:
1477:
1470:
1464:
1459:
1455:
1450:
1444:
1440:
1436:
1432:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1386:
1378:
1376:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1311:
1269:
1262:
1257:
1254:
1247:
1242:
1238:
1234:
1230:
1226:
1222:
1216:
1212:
1208:
1204:
1199:
1193:
1186:
1181:
1179:
1152:
1135:
1131:
1127:
1123:
1097:
891:January 2022
888:
884:adding to it
879:
863:Non-examples
855:
851:
847:
843:
839:
835:
831:
827:
823:
819:
816:localization
811:
807:
800:
781:The ring of
770:
765:
753:
749:
745:
741:
730:
726:
722:
715:
711:
708:dual numbers
336:
266:local domain
265:
264:is called a
259:
254:
244:
235:
228:
224:
219:
212:
205:
193:
187:
177:
171:
167:
163:
156:
145:
131:
124:
120:
115:
101:
98:Stellenringe
97:
91:
80:
63:
62:, or prime.
42:are certain
39:
29:
2893:Ring theory
2627:for such a
2485:projective
2407:factor ring
1829:belongs to
1582:, then the
1546:has finite
1316:of a field
805:prime ideal
283:skew fields
218:are called
87:prime ideal
72:commutative
40:local rings
36:ring theory
32:mathematics
2887:Categories
2794:References
2775:"Tag 07BI"
2760:2024-08-26
2418:skew field
2385:is not in
1912:completion
1659:Noetherian
1589:is local.
1524:direct sum
1237:such that
1169:, and the
762:invertible
616:such that
251:Noetherian
181:is a unit.
176:1 −
125:local ring
104:is due to
102:local ring
2801:Lam, T.Y.
2680:115691729
2279:→
2163:↦
2070:→
1798:−
1735:with the
1688:∞
1673:⋂
1163:real line
986:−
791:localized
758:nilpotent
638:∑
582:∞
567:∑
535:∞
520:∑
488:∞
473:∑
427:∞
412:∑
366:The ring
350:nilpotent
288:The ring
202:principal
52:manifolds
2845:See also
2825:(2009).
2803:(2001).
2152:sending
1996: :
1904:complete
1856:for any
1505:value at
1122:, where
803:and any
272:Examples
200:proper (
54:, or of
2727:1990215
1528:modules
1292:schemes
1286:on any
1265:(0) = 0
1250:(0) ≠ 0
1196:(0) ≠ 0
1189:(0) ≠ 0
1134:) and (
704:algebra
334:prime,
220:coprime
198:coprime
170:, then
136:maximal
106:Zariski
76:modules
2833:
2811:
2725:
2678:
1906:(as a
1580:-group
1569:> 0
1548:length
1465:is an
1447:) = 0,
1340:is in
1211:) = 1/
814:, the
739:modulo
279:fields
2723:JSTOR
2701:(PDF)
2676:S2CID
2643:Notes
2481:is a
2420:. If
2416:is a
1988:is a
1980:from
1645:. If
1630:as a
1559:field
1557:is a
1428:with
1348:with
1260:with
1175:germs
838:with
793:at 2.
785:with
744:. If
281:(and
153:units
140:ideal
138:left
123:is a
85:at a
60:place
44:rings
2831:ISBN
2809:ISBN
2664:1938
2467:free
2334:The
2023:and
2014:) ⊆
1964:and
1571:and
1439:) =
1420:and
1336:and
1126:and
975:and
903:The
846:and
277:All
118:ring
2713:doi
2668:doi
2457:by
2393:If
2210:is
1984:to
1952:If
1902:is
1641:on
1561:of
1553:If
1538:is
1469:at
1400:of
1350:odd
1328:of
1252:.)
1241:(0)
886:.
822:at
818:of
810:of
787:odd
773:= 2
760:or
706:of
339:≥ 1
222:if
174:or
155:in
50:or
30:In
2889::
2753:.
2721:.
2709:53
2707:.
2703:.
2674:.
2639:.
2453:A
2450:.
2425:≠
2399:,
2326:.
2000:→
1970:,
1958:,
1949:.
1934:/
1929:,
1918:.
1896:,
1651:,
1608:,
1587:kG
1497:tX
1495:=
1374:.
1298:.
1267:.
1227:fg
1098:pq
1096:/(
854:-
850:∈
842:∈
775:.)
752:/(
725:/(
268:.
242:.
234:+
227:=
211:,
116:A
108:.
89:.
78:.
38:,
2839:.
2817:.
2777:.
2763:.
2729:.
2715::
2682:.
2670::
2637:R
2633:R
2629:P
2615:)
2612:P
2609:(
2604:R
2599:d
2596:n
2593:E
2581:R
2567:)
2564:R
2561:(
2556:n
2551:M
2529:)
2526:P
2523:(
2518:R
2513:d
2510:n
2507:E
2495:R
2491:P
2487:R
2479:P
2448:J
2446:/
2444:m
2440:J
2438:/
2436:R
2432:R
2427:R
2423:J
2414:m
2412:/
2410:R
2403:)
2401:m
2397:R
2395:(
2389:.
2387:m
2383:x
2377:x
2371:x
2365:x
2358:R
2354:x
2347:R
2343:R
2339:m
2314:)
2309:2
2305:x
2301:(
2297:/
2293:]
2290:x
2287:[
2283:C
2276:)
2271:3
2267:x
2263:(
2259:/
2255:]
2252:x
2249:[
2245:C
2224:)
2221:x
2218:(
2198:)
2195:y
2192:,
2189:x
2186:(
2166:x
2160:x
2140:)
2135:4
2131:y
2127:,
2124:y
2119:2
2115:x
2111:,
2106:3
2102:x
2098:(
2094:/
2090:]
2087:y
2084:,
2081:x
2078:[
2074:C
2067:)
2062:3
2058:x
2054:(
2050:/
2046:]
2043:x
2040:[
2036:C
2025:S
2021:R
2016:n
2012:m
2010:(
2008:f
2002:S
1998:R
1994:f
1986:S
1982:R
1974:)
1972:n
1968:S
1966:(
1962:)
1960:m
1956:R
1954:(
1947:P
1936:m
1932:R
1927:P
1923:R
1900:)
1898:m
1894:R
1892:(
1873:n
1869:x
1858:n
1842:n
1838:m
1811:2
1807:x
1803:1
1794:e
1773:)
1770:x
1767:(
1757:m
1753:R
1737:m
1733:R
1727:(
1712:}
1709:0
1706:{
1703:=
1698:i
1694:m
1683:1
1680:=
1677:i
1655:)
1653:m
1649:R
1647:(
1643:R
1637:m
1628:m
1620:m
1616:R
1612:)
1610:m
1606:R
1604:(
1578:p
1573:G
1567:p
1555:k
1544:M
1536:M
1532:M
1508:P
1499:,
1493:Y
1484:,
1482:X
1480:/
1478:Y
1471:P
1460:G
1458:/
1456:F
1445:P
1443:(
1441:G
1437:P
1435:(
1433:F
1426:V
1422:G
1418:F
1414:V
1410:P
1406:R
1402:V
1398:P
1394:V
1387:K
1379:K
1361:Q
1342:R
1338:x
1334:x
1330:K
1326:x
1322:R
1318:K
1263:f
1258:f
1248:f
1243:g
1239:f
1235:g
1231:f
1223:g
1219:)
1217:x
1215:(
1213:f
1209:x
1207:(
1205:g
1200:f
1194:f
1187:f
1182:f
1136:q
1132:p
1128:q
1124:p
1109:Z
1100:)
1083:Z
1072:.
1060:p
1040:)
1037:p
1034:(
1013:Z
989:x
983:1
963:x
943:K
923:]
920:x
917:[
914:K
893:)
889:(
858:.
856:P
852:R
848:s
844:P
840:a
836:s
834:/
832:a
828:P
824:P
820:R
812:R
808:P
801:R
771:n
766:F
754:X
750:F
746:F
742:X
731:F
727:X
723:F
716:n
712:F
674:j
670:b
664:i
660:a
654:n
651:=
648:j
645:+
642:i
634:=
629:n
625:c
602:i
598:x
592:i
588:c
577:0
574:=
571:i
563:=
560:)
555:i
551:x
545:i
541:b
530:0
527:=
524:i
516:(
513:)
508:i
504:x
498:i
494:a
483:0
480:=
477:i
469:(
447:i
443:x
437:i
433:a
422:0
419:=
416:i
391:]
388:]
385:x
382:[
379:[
375:C
345:.
343:p
337:n
332:p
317:Z
311:n
307:p
302:/
297:Z
239:2
236:I
232:1
229:I
225:R
216:2
213:I
209:1
206:I
194:R
178:x
172:x
168:R
164:x
157:R
146:R
142:.
132:R
121:R
20:)
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