2947:
1884:
1767:
115:
2845:
2908:
1294:
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.
2381:
1509:
2751:
2322:
959:
750:
176:
688:
1879:{\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\}.}
834:
3134:
3102:
3078:
3050:
1465:
3599:
1714:
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
986:
885:
569:
2933:, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary.
2608:
2585:
2537:
2517:
2493:
2473:
2453:
2424:
2404:
2259:
2235:
2203:
2170:
2150:
2126:
2106:
2078:
2055:
2032:
1980:
1122:
1059:
858:
621:, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa.
3175:
says that, over a left
Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the
3753:
3723:
3647:
3550:
3530:
3384:
61:
2756:
3371:. Mathematics and Its Applications. Soviet Series. Vol. 70. Translated by Bakhturin, Yu. A. Dordrecht: Kluwer Academic Publishers.
2926:
2861:
562:
3580:
3509:
1257:
1320:(left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of
2929:, already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and (Noetherian)
1178:
3692:
2328:
1478:
3779:
3493:
1674:
is a subring of a field, any integral domain that is not
Noetherian provides an example. To give a less trivial example,
555:
180:
Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is
2686:
1287:
3774:
1976:
1530:
Rings that are not
Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings:
1371:
424:
1604:
is not
Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (2), (2), (2), ...
235:
212:
3594:
1983:. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain.
3745:
3105:
2922:
2915:
2273:
1663:
1336:
1062:
231:
216:
204:
3176:
3246:
2930:
2636:
893:
43:
1990:
is not
Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in
1937:
1214:
2241:
This is because there is a bijection between the left and right ideals of the group ring in this case, via the
181:
697:
515:
126:
1468:
1404:
1280:
1066:
2858:, on the other hand, gives some information about a descending chain of ideals given by powers of ideals
647:
3019:
2632:
1520:
1403:, such as the integers, is Noetherian since every ideal is generated by a single element. This includes
1321:
1313:
793:
2855:
50:; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said
3796:
3769:
3525:
Atiyah, M. F., MacDonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley-Longman.
3172:
3115:
3083:
3059:
3031:
1446:
1400:
1355:
1302:
1193:
502:
494:
466:
461:
452:
409:
351:
2978:
2848:
2644:
2262:
2085:
2035:
1608:
1347:
1332:
1181:
over a commutative
Noetherian ring is Noetherian. (This follows from the two previous properties.)
1164:
1132:
641:
588:
520:
510:
361:
261:
253:
244:
195:
ring theory since many rings that are encountered in mathematics are
Noetherian (in particular the
192:
47:
3618:
3590:
2911:
2081:
2058:
1991:
1363:
888:
326:
317:
275:
39:
211:), and many general theorems on rings rely heavily on the Noetherian property (for example, the
3679:. Graduate Texts in Mathematics. Vol. 131 (2nd ed.). New York: Springer. p. 19.
3749:
3719:
3688:
3643:
3576:
3546:
3526:
3505:
3390:
3380:
3263:
3188:
3164:
1679:
1601:
1526:
The ring of polynomials in finitely-many variables over the integers or a field is
Noetherian.
1200:
1171:
764:
2925:
of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem,
3729:
3680:
3661:
3635:
3608:
3568:
3536:
3497:
3406:
3372:
3255:
2990:
2619:
2588:
2551:
2544:
2496:
2129:
1419:
1408:
1317:
618:
346:
196:
188:
3702:
3657:
3519:
3402:
964:
863:
3733:
3698:
3665:
3653:
3515:
3410:
3398:
3026:
2540:
1671:
1426:
1415:
1385:
1207:
1030:
1009:
768:
438:
432:
419:
399:
390:
356:
293:
200:
3560:
2593:
2561:
2547:
2522:
2502:
2478:
2458:
2429:
2409:
2389:
2244:
2211:
2179:
2155:
2135:
2111:
2091:
2063:
2040:
2008:
1987:
1430:
1393:
1189:
1072:
1035:
843:
480:
2946:
3790:
3672:
3193:
2993:
over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring
2640:
1298:
1291:
1151:
1001:
783:
366:
331:
288:
227:
2652:
2610:
whose group ring over any Noetherian commutative ring is not two-sided Noetherian.
2555:
2265:
1516:
1422:) is a Noetherian domain in which every ideal is generated by at most two elements.
1269:
540:
471:
305:
223:
208:
3259:
3241:
3540:
1000:
of the ring is finitely generated. However, it is not enough to ask that all the
17:
3237:
3109:
1670:
However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any
1512:
1441:
1389:
997:
530:
525:
414:
404:
378:
371:
31:
3711:
3684:
3639:
3572:
3501:
3376:
3168:
2003:
1979:
is not necessarily a Noetherian ring. It does satisfy a weaker condition: the
1726:
1467:
is a both left and right Noetherian ring; this follows from the fact that the
1306:
1168:
1013:
1005:
280:
3394:
3267:
3152:
756:
535:
341:
298:
266:
3343:
1276:
is a Noetherian ring (see the "faithfully flat" article for the reasoning).
117:
of left (or right) ideals has a largest element; that is, there exists an
2854:
A Noetherian ring is defined in terms of ascending chains of ideals. The
1719:
336:
3496:, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376,
3622:
1222:
1396:, is Noetherian. (A field only has two ideals — itself and (0).)
591:, it is necessary to distinguish between three very similar concepts:
270:
3613:
1433:
is a Noetherian ring, as a consequence of the Hilbert basis theorem.
110:{\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots }
2989:
Given a ring, there is a close connection between the behaviors of
2840:{\displaystyle (f)=(p_{1}^{n_{1}})\cap \cdots \cap (p_{r}^{n_{r}})}
1948:
of the previous paragraph is a subring of the left Noetherian ring
1889:
This ring is right Noetherian, but not left Noetherian; the subset
778:
The following condition is also an equivalent condition for a ring
1611:
from the real numbers to the real numbers is not Noetherian: Let
1359:
3567:. Graduate Texts in Mathematics. Vol. 150. Springer-Verlag.
2847:
and thus the primary decomposition is a direct generalization of
2618:
Many important theorems in ring theory (especially the theory of
184:. A ring is Noetherian if it is both left- and right-Noetherian.
2903:{\displaystyle I\supseteq I^{2}\supseteq I^{3}\supseteq \cdots }
2587:
is two-sided Noetherian. On the other hand, however, there is a
3344:
The ring of stable homotopy groups of spheres is not noetherian
1331:
In a commutative Noetherian ring, there are only finitely many
996:
For a commutative ring to be Noetherian it suffices that every
3630:
Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki (2008),
2941:
1312:(Bass) A ring is (left/right) Noetherian if and only if every
1210:
Noetherian module over it, then the ring is a Noetherian ring.
606:
if it satisfies the ascending chain condition on right ideals.
226:, but the importance of the concept was recognized earlier by
1905:= 0 is a left ideal that is not finitely generated as a left
599:
if it satisfies the ascending chain condition on left ideals.
3744:, Cambridge Studies in Advanced Mathematics (2nd ed.),
1783:
3565:
Commutative Algebra with a View Toward Algebraic Geometry
2622:) rely on the assumptions that the rings are Noetherian.
1012:(see a counterexample to Krull's intersection theorem at
234:(which asserts that polynomial rings are Noetherian) and
2753:
is a product of powers of distinct prime elements, then
2958:
2376:{\displaystyle g\mapsto g^{-1}\qquad (\forall g\in G).}
3634:, Progress in Mathematics, vol. 236, BirkhÀuser,
3632:
D-modules, perverse sheaves, and representation theory
3322:, Ch III, §2, no. 10, Remarks at the end of the number
1791:
1659:, etc., is an ascending chain that does not terminate.
1534:
The ring of polynomials in infinitely-many variables,
1358:). Thus, if, in addition, the factorization is unique
896:
3331:
3242:"Commutative rings with restricted minimum condition"
3118:
3086:
3062:
3034:
2864:
2759:
2689:
2631:
Over a commutative Noetherian ring, each ideal has a
2596:
2564:
2525:
2505:
2481:
2461:
2432:
2412:
2392:
2331:
2276:
2247:
2214:
2182:
2158:
2138:
2114:
2094:
2066:
2043:
2011:
1770:
1504:{\displaystyle \operatorname {Sym} ({\mathfrak {g}})}
1481:
1449:
1325:
1075:
1038:
1004:
are finitely generated, as there is a non-Noetherian
967:
866:
846:
796:
700:
650:
129:
64:
624:
There are other, equivalent, definitions for a ring
3354:
2746:{\displaystyle f=p_{1}^{n_{1}}\cdots p_{r}^{n_{r}}}
1916:is a commutative subring of a left Noetherian ring
3718:(Third ed.), Reading, Mass.: Addison-Wesley,
3128:
3096:
3072:
3044:
2902:
2839:
2745:
2602:
2579:
2531:
2511:
2487:
2467:
2447:
2418:
2398:
2375:
2316:
2253:
2229:
2197:
2164:
2144:
2120:
2100:
2072:
2049:
2026:
1878:
1757:. Choosing a basis, we can describe the same ring
1503:
1459:
1116:
1053:
980:
953:
879:
852:
828:
744:
682:
170:
109:
775:Similar results hold for right-Noetherian rings.
58:respectively. That is, every increasing sequence
3600:Proceedings of the American Mathematical Society
3303:
3301:
2172:, the following two conditions are equivalent.
1511:, which is a polynomial ring over a field (the
1283:of a commutative Noetherian ring is Noetherian.
3471:
3459:
3447:
3435:
3307:
1515:); thus, Noetherian. For the same reason, the
1248:is a subring of a commutative Noetherian ring
2317:{\displaystyle R\to R^{\operatorname {op} },}
1981:ascending chain condition on principal ideals
1597:), etc. is ascending, and does not terminate.
563:
8:
3488:Anderson, Frank W.; Fuller, Kent R. (1992),
1163:is also Noetherian. Stated differently, the
954:{\textstyle f_{i}=\sum _{j=1}^{n}r_{j}f_{j}}
3052:such that each injective left module over
570:
556:
240:
27:Mathematical ring with well-behaved ideals
3612:
3367:OlâshanskiÄ, Aleksandr Yurâevich (1991).
3292:
3280:
3167:of an indecomposable injective module is
3120:
3119:
3117:
3088:
3087:
3085:
3064:
3063:
3061:
3036:
3035:
3033:
3007:(Bass) Each direct sum of injective left
2910:. It is a technical tool that is used to
2888:
2875:
2863:
2826:
2821:
2816:
2789:
2784:
2779:
2758:
2735:
2730:
2725:
2710:
2705:
2700:
2688:
2595:
2563:
2524:
2504:
2480:
2460:
2455:is left/right/two-sided Noetherian, then
2431:
2411:
2391:
2342:
2330:
2305:
2275:
2246:
2213:
2181:
2157:
2137:
2113:
2093:
2065:
2042:
2010:
1932:is Noetherian. (In the special case when
1863:
1849:
1835:
1828:
1822:
1786:
1769:
1618:be the ideal of all continuous functions
1492:
1491:
1480:
1451:
1450:
1448:
1105:
1086:
1074:
1037:
972:
966:
945:
935:
925:
914:
901:
895:
871:
865:
845:
814:
801:
795:
736:
714:
699:
674:
655:
649:
613:if it is both left- and right-Noetherian.
187:Noetherian rings are fundamental in both
147:
134:
128:
95:
82:
69:
63:
3423:
3369:Geometry of defining relations in groups
3319:
3224:
3205:
2635:, meaning that it can be written as an
2475:is left/right/two-sided Noetherian and
1346:, every element can be factorized into
745:{\displaystyle I=Ra_{1}+\cdots +Ra_{n}}
243:
3677:A first course in noncommutative rings
171:{\displaystyle I_{n}=I_{n+1}=\cdots .}
2519:is a Noetherian commutative ring and
7:
3593:; Jategaonkar, Arun Vinayak (1974).
3332:Hotta, Takeuchi & Tanisaki (2008
3211:
3209:
1233:is a finitely generated module over
3121:
3089:
3065:
3037:
1493:
1452:
1384:Any field, including the fields of
1342:In a commutative Noetherian domain
1174:of a Noetherian ring is Noetherian.
683:{\displaystyle a_{1},\ldots ,a_{n}}
2355:
829:{\displaystyle f_{1},f_{2},\dots }
25:
3542:Commutative Algebra: Chapters 1-7
2647:are all distinct) where an ideal
1936:is commutative, this is known as
1664:stable homotopy groups of spheres
1362:multiplication of the factors by
1326:#Implication on injective modules
1225:of a commutative Noetherian ring
222:Noetherian rings are named after
3080:-generated modules (a module is
2997:, the following are equivalent:
2985:Implication on injective modules
2945:
1964:is finitely generated as a left
1940:.) However, this is not true if
1924:is finitely generated as a left
1864:
1850:
1836:
1555:, etc. The sequence of ideals (
1288:AkizukiâHopkinsâLevitzki theorem
782:to be left-Noetherian and it is
3490:Rings and categories of modules
3355:Formanek & Jategaonkar 1974
3155:into a direct sum of copies of
3129:{\displaystyle {\mathfrak {c}}}
3097:{\displaystyle {\mathfrak {c}}}
3073:{\displaystyle {\mathfrak {c}}}
3045:{\displaystyle {\mathfrak {c}}}
2927:Krull's principal ideal theorem
2914:other key theorems such as the
2351:
1460:{\displaystyle {\mathfrak {g}}}
1297:A left Noetherian ring is left
1206:If a commutative ring admits a
1150:is a two-sided ideal, then the
1029:is a Noetherian ring, then the
3595:"Subrings of Noetherian rings"
3025:(FaithâWalker) There exists a
2834:
2809:
2797:
2772:
2766:
2760:
2574:
2568:
2442:
2436:
2367:
2352:
2335:
2302:
2295:
2289:
2286:
2280:
2224:
2218:
2192:
2186:
2021:
2015:
1498:
1488:
1192:every finitely generated left
1111:
1079:
1048:
1042:
1:
3494:Graduate Texts in Mathematics
3260:10.1215/S0012-7094-50-01704-2
2683:. For example, if an element
1944:is not commutative: the ring
1737:be the ring of homomorphisms
3740:Matsumura, Hideyuki (1989),
2851:of integers and polynomials.
1897:consisting of elements with
1519:, and more general rings of
1324:injective modules. See also
1264:(or more generally exhibits
1124:is a Noetherian ring. Also,
644:, i.e. there exist elements
3775:Encyclopedia of Mathematics
3018:-module is a direct sum of
2651:is called primary if it is
1977:unique factorization domain
1372:unique factorization domain
1014:Local ring#Commutative case
205:rings of algebraic integers
3813:
3746:Cambridge University Press
3472:Anderson & Fuller 1992
3460:Anderson & Fuller 1992
3448:Anderson & Fuller 1992
3436:Anderson & Fuller 1992
3334:, §D.1, Proposition 1.4.6)
3308:Anderson & Fuller 1992
3004:is a left Noetherian ring.
2931:universally catenary rings
2916:Krull intersection theorem
2679:for some positive integer
1698:is a subring of the field
1638:. The sequence of ideals
1337:descending chain condition
840:, there exists an integer
217:Krull intersection theorem
3685:10.1007/978-1-4419-8616-0
3640:10.1007/978-0-8176-4523-6
3573:10.1007/978-1-4612-5350-1
3502:10.1007/978-1-4612-4418-9
3377:10.1007/978-94-011-3618-1
3247:Duke Mathematical Journal
1177:Every finitely-generated
1144:is a Noetherian ring and
786:'s original formulation:
44:ascending chain condition
1972:is not left Noetherian.
1710:) in only two variables.
1440:of a finite-dimensional
236:Hilbert's syzygy theorem
3742:Commutative Ring Theory
3104:-generated if it has a
2152:and a commutative ring
1994:but is not Noetherian.
1436:The enveloping algebra
1405:principal ideal domains
1135:, is a Noetherian ring.
1063:Hilbert's basis theorem
1008:whose maximal ideal is
628:to be left-Noetherian:
232:Hilbert's basis theorem
3130:
3098:
3074:
3046:
3011:-modules is injective.
2904:
2841:
2747:
2604:
2581:
2533:
2513:
2489:
2469:
2449:
2420:
2400:
2377:
2318:
2255:
2231:
2199:
2166:
2146:
2122:
2102:
2074:
2051:
2028:
1998:Noetherian group rings
1880:
1521:differential operators
1505:
1469:associated graded ring
1461:
1339:holds on prime ideals.
1301:and a left Noetherian
1118:
1055:
982:
955:
930:
881:
854:
830:
759:set of left ideals of
746:
684:
213:LaskerâNoether theorem
172:
111:
3177:KrullâSchmidt theorem
3147:such that every left
3131:
3099:
3075:
3047:
2905:
2842:
2748:
2633:primary decomposition
2605:
2582:
2534:
2514:
2490:
2470:
2450:
2421:
2401:
2378:
2319:
2256:
2232:
2200:
2167:
2147:
2123:
2103:
2075:
2052:
2029:
1881:
1745:to itself satisfying
1506:
1462:
1286:A consequence of the
1244:Similarly, if a ring
1241:is a Noetherian ring.
1119:
1061:is Noetherian by the
1056:
983:
981:{\displaystyle r_{j}}
956:
910:
882:
880:{\displaystyle f_{i}}
855:
831:
747:
685:
173:
112:
3310:, Proposition 18.13.
3139:There exists a left
3116:
3084:
3060:
3032:
3014:Each injective left
2938:Non-commutative case
2862:
2757:
2687:
2594:
2562:
2523:
2503:
2479:
2459:
2430:
2410:
2390:
2329:
2274:
2245:
2237:is right-Noetherian.
2212:
2180:
2156:
2136:
2112:
2092:
2064:
2041:
2009:
1768:
1609:continuous functions
1479:
1447:
1401:principal ideal ring
1356:factorization domain
1348:irreducible elements
1333:minimal prime ideals
1073:
1036:
965:
894:
864:
844:
794:
767:by inclusion, has a
698:
648:
589:noncommutative rings
467:Group with operators
410:Complemented lattice
245:Algebraic structures
230:, with the proof of
127:
62:
3545:. Springer-Verlag.
3438:, Theorem 25.6. (b)
3426:, Proposition 3.11.
3056:is a direct sum of
2849:prime factorization
2833:
2796:
2742:
2717:
2263:associative algebra
2205:is left-Noetherian.
2086:associative algebra
1290:is that every left
1188:is left-Noetherian
1179:commutative algebra
521:Composition algebra
281:Quasigroup and loop
42:that satisfies the
3126:
3094:
3070:
3042:
3022:injective modules.
2957:. You can help by
2900:
2837:
2812:
2775:
2743:
2721:
2696:
2600:
2577:
2529:
2509:
2485:
2465:
2445:
2416:
2396:
2373:
2314:
2251:
2227:
2195:
2162:
2142:
2118:
2098:
2070:
2047:
2024:
1992:algebraic geometry
1876:
1816:
1680:rational functions
1666:is not Noetherian.
1602:algebraic integers
1501:
1457:
1114:
1051:
978:
961:with coefficients
951:
889:linear combination
877:
850:
826:
742:
680:
642:finitely generated
182:finitely generated
168:
107:
46:on left and right
3770:"Noetherian ring"
3755:978-0-521-36764-6
3725:978-0-201-55540-0
3649:978-0-8176-4363-8
3552:978-0-387-19371-7
3537:Bourbaki, Nicolas
3531:978-0-201-40751-8
3462:, Corollary 26.3.
3386:978-0-7923-1394-6
3215:Lam (2001), p. 19
3189:Noetherian scheme
3173:Azumaya's theorem
3165:endomorphism ring
2991:injective modules
2975:
2974:
2639:of finitely many
2620:commutative rings
2603:{\displaystyle G}
2580:{\displaystyle R}
2532:{\displaystyle G}
2512:{\displaystyle R}
2499:. Conversely, if
2488:{\displaystyle G}
2468:{\displaystyle R}
2448:{\displaystyle R}
2419:{\displaystyle R}
2399:{\displaystyle G}
2254:{\displaystyle R}
2230:{\displaystyle R}
2198:{\displaystyle R}
2165:{\displaystyle R}
2145:{\displaystyle G}
2121:{\displaystyle R}
2101:{\displaystyle R}
2073:{\displaystyle R}
2050:{\displaystyle G}
2027:{\displaystyle R}
1523:, are Noetherian.
1475:is a quotient of
1420:rings of integers
1409:Euclidean domains
1201:Noetherian module
1172:ring homomorphism
1133:power series ring
1117:{\displaystyle R}
1054:{\displaystyle R}
853:{\displaystyle n}
790:Given a sequence
765:partially ordered
632:Every left ideal
619:commutative rings
583:Characterizations
580:
579:
18:Noetherian domain
16:(Redirected from
3804:
3783:
3758:
3736:
3706:
3668:
3626:
3616:
3591:Formanek, Edward
3586:
3556:
3522:
3475:
3469:
3463:
3457:
3451:
3445:
3439:
3433:
3427:
3421:
3415:
3414:
3364:
3358:
3352:
3346:
3341:
3335:
3329:
3323:
3317:
3311:
3305:
3296:
3290:
3284:
3278:
3272:
3271:
3234:
3228:
3222:
3216:
3213:
3135:
3133:
3132:
3127:
3125:
3124:
3103:
3101:
3100:
3095:
3093:
3092:
3079:
3077:
3076:
3071:
3069:
3068:
3051:
3049:
3048:
3043:
3041:
3040:
2979:Goldie's theorem
2970:
2967:
2949:
2942:
2923:dimension theory
2909:
2907:
2906:
2901:
2893:
2892:
2880:
2879:
2856:ArtinâRees lemma
2846:
2844:
2843:
2838:
2832:
2831:
2830:
2820:
2795:
2794:
2793:
2783:
2752:
2750:
2749:
2744:
2741:
2740:
2739:
2729:
2716:
2715:
2714:
2704:
2626:Commutative case
2609:
2607:
2606:
2601:
2589:Noetherian group
2586:
2584:
2583:
2578:
2552:polycyclic group
2538:
2536:
2535:
2530:
2518:
2516:
2515:
2510:
2497:Noetherian group
2494:
2492:
2491:
2486:
2474:
2472:
2471:
2466:
2454:
2452:
2451:
2446:
2425:
2423:
2422:
2417:
2405:
2403:
2402:
2397:
2382:
2380:
2379:
2374:
2350:
2349:
2323:
2321:
2320:
2315:
2310:
2309:
2260:
2258:
2257:
2252:
2236:
2234:
2233:
2228:
2204:
2202:
2201:
2196:
2171:
2169:
2168:
2163:
2151:
2149:
2148:
2143:
2127:
2125:
2124:
2119:
2107:
2105:
2104:
2099:
2079:
2077:
2076:
2071:
2056:
2054:
2053:
2048:
2033:
2031:
2030:
2025:
1885:
1883:
1882:
1877:
1872:
1868:
1867:
1853:
1839:
1827:
1823:
1821:
1820:
1600:The ring of all
1510:
1508:
1507:
1502:
1497:
1496:
1466:
1464:
1463:
1458:
1456:
1455:
1386:rational numbers
1162:
1149:
1143:
1130:
1123:
1121:
1120:
1115:
1110:
1109:
1091:
1090:
1060:
1058:
1057:
1052:
987:
985:
984:
979:
977:
976:
960:
958:
957:
952:
950:
949:
940:
939:
929:
924:
906:
905:
886:
884:
883:
878:
876:
875:
859:
857:
856:
851:
835:
833:
832:
827:
819:
818:
806:
805:
751:
749:
748:
743:
741:
740:
719:
718:
689:
687:
686:
681:
679:
678:
660:
659:
604:right-Noetherian
572:
565:
558:
347:Commutative ring
276:Rack and quandle
241:
201:polynomial rings
197:ring of integers
177:
175:
174:
169:
158:
157:
139:
138:
122:
116:
114:
113:
108:
100:
99:
87:
86:
74:
73:
56:right-Noetherian
21:
3812:
3811:
3807:
3806:
3805:
3803:
3802:
3801:
3787:
3786:
3768:
3765:
3756:
3739:
3726:
3710:
3695:
3671:
3650:
3629:
3614:10.2307/2039890
3589:
3583:
3561:Eisenbud, David
3559:
3553:
3535:
3512:
3487:
3484:
3479:
3478:
3470:
3466:
3458:
3454:
3450:, Theorem 25.8.
3446:
3442:
3434:
3430:
3422:
3418:
3387:
3366:
3365:
3361:
3353:
3349:
3342:
3338:
3330:
3326:
3318:
3314:
3306:
3299:
3291:
3287:
3279:
3275:
3238:Cohen, Irvin S.
3236:
3235:
3231:
3227:, Exercise 1.1.
3223:
3219:
3214:
3207:
3202:
3185:
3114:
3113:
3082:
3081:
3058:
3057:
3030:
3029:
3027:cardinal number
2987:
2971:
2965:
2962:
2955:needs expansion
2940:
2884:
2871:
2860:
2859:
2822:
2785:
2755:
2754:
2731:
2706:
2685:
2684:
2628:
2616:
2592:
2591:
2560:
2559:
2521:
2520:
2501:
2500:
2477:
2476:
2457:
2456:
2428:
2427:
2408:
2407:
2406:be a group and
2388:
2387:
2338:
2327:
2326:
2301:
2272:
2271:
2243:
2242:
2210:
2209:
2178:
2177:
2154:
2153:
2134:
2133:
2110:
2109:
2090:
2089:
2062:
2061:
2039:
2038:
2007:
2006:
2000:
1938:Eakin's theorem
1815:
1814:
1809:
1803:
1802:
1797:
1787:
1785:
1782:
1781:
1777:
1766:
1765:
1672:integral domain
1658:
1651:
1644:
1616:
1596:
1589:
1582:
1575:
1568:
1561:
1554:
1547:
1540:
1477:
1476:
1445:
1444:
1427:coordinate ring
1416:Dedekind domain
1394:complex numbers
1381:
1258:faithfully flat
1154:
1145:
1139:
1125:
1101:
1082:
1071:
1070:
1034:
1033:
1031:polynomial ring
1022:
968:
963:
962:
941:
931:
897:
892:
891:
867:
862:
861:
860:such that each
842:
841:
836:of elements in
810:
797:
792:
791:
769:maximal element
732:
710:
696:
695:
670:
651:
646:
645:
597:left-Noetherian
585:
576:
547:
546:
545:
516:Non-associative
498:
487:
486:
476:
456:
445:
444:
433:Map of lattices
429:
425:Boolean algebra
420:Heyting algebra
394:
383:
382:
376:
357:Integral domain
321:
310:
309:
303:
257:
143:
130:
125:
124:
118:
91:
78:
65:
60:
59:
52:left-Noetherian
36:Noetherian ring
28:
23:
22:
15:
12:
11:
5:
3810:
3808:
3800:
3799:
3789:
3788:
3785:
3784:
3764:
3763:External links
3761:
3760:
3759:
3754:
3737:
3724:
3707:
3693:
3673:Lam, Tsit Yuen
3669:
3648:
3627:
3607:(2): 181â186.
3587:
3581:
3557:
3551:
3533:
3523:
3510:
3483:
3480:
3477:
3476:
3464:
3452:
3440:
3428:
3416:
3385:
3359:
3347:
3336:
3324:
3312:
3297:
3295:, Theorem 3.6.
3293:Matsumura 1989
3285:
3283:, Theorem 3.5.
3281:Matsumura 1989
3273:
3229:
3217:
3204:
3203:
3201:
3198:
3197:
3196:
3191:
3184:
3181:
3161:
3160:
3137:
3123:
3106:generating set
3091:
3067:
3039:
3023:
3020:indecomposable
3012:
3005:
2986:
2983:
2982:
2981:
2973:
2972:
2952:
2950:
2939:
2936:
2935:
2934:
2919:
2899:
2896:
2891:
2887:
2883:
2878:
2874:
2870:
2867:
2852:
2836:
2829:
2825:
2819:
2815:
2811:
2808:
2805:
2802:
2799:
2792:
2788:
2782:
2778:
2774:
2771:
2768:
2765:
2762:
2738:
2734:
2728:
2724:
2720:
2713:
2709:
2703:
2699:
2695:
2692:
2641:primary ideals
2627:
2624:
2615:
2612:
2599:
2576:
2573:
2570:
2567:
2548:solvable group
2528:
2508:
2484:
2464:
2444:
2441:
2438:
2435:
2415:
2395:
2384:
2383:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2348:
2345:
2341:
2337:
2334:
2324:
2313:
2308:
2304:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2250:
2239:
2238:
2226:
2223:
2220:
2217:
2206:
2194:
2191:
2188:
2185:
2161:
2141:
2132:. For a group
2117:
2097:
2069:
2046:
2023:
2020:
2017:
2014:
1999:
1996:
1988:valuation ring
1928:-module, then
1887:
1886:
1875:
1871:
1866:
1862:
1859:
1856:
1852:
1848:
1845:
1842:
1838:
1834:
1831:
1826:
1819:
1813:
1810:
1808:
1805:
1804:
1801:
1798:
1796:
1793:
1792:
1790:
1784:
1780:
1776:
1773:
1712:
1711:
1668:
1667:
1660:
1656:
1649:
1642:
1630:) = 0 for all
1614:
1605:
1598:
1594:
1587:
1580:
1573:
1566:
1559:
1552:
1545:
1538:
1528:
1527:
1524:
1500:
1495:
1490:
1487:
1484:
1454:
1434:
1431:affine variety
1423:
1412:
1397:
1380:
1377:
1376:
1375:
1340:
1329:
1322:indecomposable
1310:
1295:
1284:
1277:
1242:
1211:
1204:
1190:if and only if
1182:
1175:
1136:
1113:
1108:
1104:
1100:
1097:
1094:
1089:
1085:
1081:
1078:
1050:
1047:
1044:
1041:
1021:
1018:
1002:maximal ideals
994:
993:
975:
971:
948:
944:
938:
934:
928:
923:
920:
917:
913:
909:
904:
900:
874:
870:
849:
825:
822:
817:
813:
809:
804:
800:
773:
772:
753:
739:
735:
731:
728:
725:
722:
717:
713:
709:
706:
703:
677:
673:
669:
666:
663:
658:
654:
615:
614:
607:
600:
584:
581:
578:
577:
575:
574:
567:
560:
552:
549:
548:
544:
543:
538:
533:
528:
523:
518:
513:
507:
506:
505:
499:
493:
492:
489:
488:
485:
484:
481:Linear algebra
475:
474:
469:
464:
458:
457:
451:
450:
447:
446:
443:
442:
439:Lattice theory
435:
428:
427:
422:
417:
412:
407:
402:
396:
395:
389:
388:
385:
384:
375:
374:
369:
364:
359:
354:
349:
344:
339:
334:
329:
323:
322:
316:
315:
312:
311:
302:
301:
296:
291:
285:
284:
283:
278:
273:
264:
258:
252:
251:
248:
247:
193:noncommutative
167:
164:
161:
156:
153:
150:
146:
142:
137:
133:
106:
103:
98:
94:
90:
85:
81:
77:
72:
68:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3809:
3798:
3795:
3794:
3792:
3781:
3777:
3776:
3771:
3767:
3766:
3762:
3757:
3751:
3747:
3743:
3738:
3735:
3731:
3727:
3721:
3717:
3713:
3709:Chapter X of
3708:
3704:
3700:
3696:
3690:
3686:
3682:
3678:
3674:
3670:
3667:
3663:
3659:
3655:
3651:
3645:
3641:
3637:
3633:
3628:
3624:
3620:
3615:
3610:
3606:
3602:
3601:
3596:
3592:
3588:
3584:
3582:0-387-94268-8
3578:
3574:
3570:
3566:
3562:
3558:
3554:
3548:
3544:
3543:
3538:
3534:
3532:
3528:
3524:
3521:
3517:
3513:
3511:0-387-97845-3
3507:
3503:
3499:
3495:
3491:
3486:
3485:
3481:
3474:, Lemma 25.4.
3473:
3468:
3465:
3461:
3456:
3453:
3449:
3444:
3441:
3437:
3432:
3429:
3425:
3424:Eisenbud 1995
3420:
3417:
3412:
3408:
3404:
3400:
3396:
3392:
3388:
3382:
3378:
3374:
3370:
3363:
3360:
3356:
3351:
3348:
3345:
3340:
3337:
3333:
3328:
3325:
3321:
3320:Bourbaki 1989
3316:
3313:
3309:
3304:
3302:
3298:
3294:
3289:
3286:
3282:
3277:
3274:
3269:
3265:
3261:
3257:
3253:
3249:
3248:
3243:
3239:
3233:
3230:
3226:
3225:Eisenbud 1995
3221:
3218:
3212:
3210:
3206:
3199:
3195:
3194:Artinian ring
3192:
3190:
3187:
3186:
3182:
3180:
3178:
3174:
3170:
3166:
3158:
3154:
3150:
3146:
3142:
3138:
3111:
3107:
3055:
3028:
3024:
3021:
3017:
3013:
3010:
3006:
3003:
3000:
2999:
2998:
2996:
2992:
2984:
2980:
2977:
2976:
2969:
2966:December 2019
2960:
2956:
2953:This section
2951:
2948:
2944:
2943:
2937:
2932:
2928:
2924:
2920:
2917:
2913:
2897:
2894:
2889:
2885:
2881:
2876:
2872:
2868:
2865:
2857:
2853:
2850:
2827:
2823:
2817:
2813:
2806:
2803:
2800:
2790:
2786:
2780:
2776:
2769:
2763:
2736:
2732:
2726:
2722:
2718:
2711:
2707:
2701:
2697:
2693:
2690:
2682:
2678:
2674:
2670:
2666:
2662:
2658:
2655:and whenever
2654:
2650:
2646:
2642:
2638:
2634:
2630:
2629:
2625:
2623:
2621:
2613:
2611:
2597:
2590:
2571:
2565:
2557:
2553:
2549:
2546:
2542:
2526:
2506:
2498:
2482:
2462:
2439:
2433:
2413:
2393:
2370:
2364:
2361:
2358:
2346:
2343:
2339:
2332:
2325:
2311:
2306:
2298:
2292:
2283:
2277:
2270:
2269:
2268:
2267:
2264:
2248:
2221:
2215:
2207:
2189:
2183:
2175:
2174:
2173:
2159:
2139:
2131:
2115:
2095:
2087:
2083:
2067:
2060:
2044:
2037:
2018:
2012:
2005:
2002:Consider the
1997:
1995:
1993:
1989:
1984:
1982:
1978:
1973:
1971:
1968:-module, but
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1919:
1915:
1910:
1908:
1904:
1900:
1896:
1892:
1873:
1869:
1860:
1857:
1854:
1846:
1843:
1840:
1832:
1829:
1824:
1817:
1811:
1806:
1799:
1794:
1788:
1778:
1774:
1771:
1764:
1763:
1762:
1760:
1756:
1752:
1748:
1744:
1740:
1736:
1732:
1728:
1725:
1721:
1717:
1709:
1705:
1701:
1697:
1694:over a field
1693:
1689:
1685:
1682:generated by
1681:
1677:
1676:
1675:
1673:
1665:
1661:
1655:
1648:
1641:
1637:
1633:
1629:
1625:
1621:
1617:
1610:
1606:
1603:
1599:
1593:
1586:
1579:
1572:
1565:
1558:
1551:
1544:
1537:
1533:
1532:
1531:
1525:
1522:
1518:
1514:
1485:
1482:
1474:
1470:
1443:
1439:
1435:
1432:
1428:
1424:
1421:
1417:
1413:
1410:
1406:
1402:
1398:
1395:
1391:
1387:
1383:
1382:
1378:
1373:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1338:
1334:
1330:
1327:
1323:
1319:
1315:
1311:
1308:
1304:
1300:
1296:
1293:
1292:Artinian ring
1289:
1285:
1282:
1278:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1209:
1205:
1202:
1198:
1196:
1191:
1187:
1183:
1180:
1176:
1173:
1170:
1166:
1161:
1157:
1153:
1152:quotient ring
1148:
1142:
1137:
1134:
1128:
1106:
1102:
1098:
1095:
1092:
1087:
1083:
1076:
1068:
1064:
1045:
1039:
1032:
1028:
1024:
1023:
1019:
1017:
1015:
1011:
1007:
1003:
999:
991:
973:
969:
946:
942:
936:
932:
926:
921:
918:
915:
911:
907:
902:
898:
890:
872:
868:
847:
839:
823:
820:
815:
811:
807:
802:
798:
789:
788:
787:
785:
781:
776:
770:
766:
762:
758:
754:
737:
733:
729:
726:
723:
720:
715:
711:
707:
704:
701:
693:
675:
671:
667:
664:
661:
656:
652:
643:
639:
635:
631:
630:
629:
627:
622:
620:
612:
608:
605:
601:
598:
594:
593:
592:
590:
582:
573:
568:
566:
561:
559:
554:
553:
551:
550:
542:
539:
537:
534:
532:
529:
527:
524:
522:
519:
517:
514:
512:
509:
508:
504:
501:
500:
496:
491:
490:
483:
482:
478:
477:
473:
470:
468:
465:
463:
460:
459:
454:
449:
448:
441:
440:
436:
434:
431:
430:
426:
423:
421:
418:
416:
413:
411:
408:
406:
403:
401:
398:
397:
392:
387:
386:
381:
380:
373:
370:
368:
367:Division ring
365:
363:
360:
358:
355:
353:
350:
348:
345:
343:
340:
338:
335:
333:
330:
328:
325:
324:
319:
314:
313:
308:
307:
300:
297:
295:
292:
290:
289:Abelian group
287:
286:
282:
279:
277:
274:
272:
268:
265:
263:
260:
259:
255:
250:
249:
246:
242:
239:
237:
233:
229:
228:David Hilbert
225:
220:
218:
214:
210:
209:number fields
206:
202:
198:
194:
190:
185:
183:
178:
165:
162:
159:
154:
151:
148:
144:
140:
135:
131:
121:
104:
101:
96:
92:
88:
83:
79:
75:
70:
66:
57:
53:
49:
45:
41:
37:
33:
19:
3773:
3741:
3715:
3676:
3631:
3604:
3598:
3564:
3541:
3489:
3467:
3455:
3443:
3431:
3419:
3368:
3362:
3350:
3339:
3327:
3315:
3288:
3276:
3254:(1): 27â42.
3251:
3245:
3232:
3220:
3162:
3156:
3148:
3144:
3140:
3053:
3015:
3008:
3001:
2994:
2988:
2963:
2959:adding to it
2954:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
2648:
2637:intersection
2617:
2614:Key theorems
2556:finite group
2385:
2266:homomorphism
2240:
2001:
1985:
1974:
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1941:
1933:
1929:
1925:
1921:
1917:
1913:
1911:
1906:
1902:
1898:
1894:
1890:
1888:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1723:
1715:
1713:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1678:The ring of
1669:
1662:The ring of
1653:
1646:
1639:
1635:
1631:
1627:
1623:
1619:
1612:
1607:The ring of
1591:
1584:
1577:
1570:
1563:
1556:
1549:
1542:
1535:
1529:
1517:Weyl algebra
1472:
1437:
1390:real numbers
1367:
1351:
1343:
1335:. Also, the
1281:localization
1273:
1270:pure subring
1265:
1261:
1253:
1249:
1245:
1238:
1234:
1230:
1226:
1218:
1217:) If a ring
1215:EakinâNagata
1194:
1185:
1159:
1155:
1146:
1140:
1126:
1026:
995:
989:
887:is a finite
837:
779:
777:
774:
760:
691:
637:
633:
625:
623:
616:
610:
603:
596:
586:
541:Hopf algebra
479:
472:Vector space
437:
377:
306:Group theory
304:
269: /
224:Emmy Noether
221:
186:
179:
119:
55:
51:
35:
29:
3797:Ring theory
3712:Lang, Serge
3357:, Theorem 3
3110:cardinality
2426:a ring. If
2130:commutative
1513:PBW theorem
1442:Lie algebra
1350:(in short,
998:prime ideal
526:Lie algebra
511:Associative
415:Total order
405:Semilattice
379:Ring theory
189:commutative
123:such that:
32:mathematics
3734:0848.13001
3694:0387951830
3666:1292.00026
3482:References
3411:0732.20019
2545:Noetherian
2080:. It is a
2004:group ring
1727:isomorphic
1622:such that
1314:direct sum
1307:Ore domain
1305:is a left
1252:such that
1229:such that
1169:surjective
1020:Properties
1006:local ring
694:such that
611:Noetherian
609:A ring is
602:A ring is
595:A ring is
3780:EMS Press
3395:0169-6378
3268:0012-7094
3171:and thus
2898:⋯
2895:⊇
2882:⊇
2869:⊇
2807:∩
2804:⋯
2801:∩
2719:⋯
2663:, either
2541:extension
2362:∈
2356:∀
2344:−
2336:↦
2290:→
2208:The ring
2176:The ring
2084:, and an
1909:-module.
1861:∈
1858:γ
1847:∈
1844:β
1833:∈
1812:γ
1800:β
1486:
1318:injective
1096:…
1067:induction
1010:principal
912:∑
824:…
757:non-empty
724:⋯
665:…
536:Bialgebra
342:Near-ring
299:Lie group
267:Semigroup
163:⋯
105:⋯
102:⊆
89:⊆
76:⊆
3791:Category
3714:(1993),
3675:(2001).
3563:(1995).
3539:(1989).
3240:(1950).
3183:See also
3151:-module
3143:-module
3112:at most
2645:radicals
2550:(i.e. a
1901:= 0 and
1720:subgroup
1690: /
1379:Examples
1299:coherent
1272:), then
1208:faithful
372:Lie ring
337:Semiring
215:and the
3782:, 2001
3716:Algebra
3703:1838439
3658:2357361
3623:2039890
3520:1245487
3403:1191619
2643:(whose
2558:, then
2554:) by a
2057:over a
1960:), and
1418:(e.g.,
1366:, then
1237:, then
1223:subring
1197:-module
1184:A ring
1167:of any
784:Hilbert
503:Algebra
495:Algebra
400:Lattice
391:Lattice
3752:
3732:
3722:
3701:
3691:
3664:
3656:
3646:
3621:
3579:
3549:
3529:
3518:
3508:
3409:
3401:
3393:
3383:
3266:
3153:embeds
2653:proper
2539:is an
1952:= Hom(
1920:, and
1733:, let
1429:of an
1392:, and
1328:below.
1303:domain
1279:Every
1131:, the
755:Every
531:Graded
462:Module
453:Module
352:Domain
271:Monoid
203:, and
48:ideals
3619:JSTOR
3200:Notes
3169:local
2912:prove
2543:of a
2495:is a
2088:over
2036:group
2034:of a
1741:from
1718:is a
1370:is a
1364:units
1360:up to
1354:is a
1268:as a
1260:over
1221:is a
1199:is a
1165:image
1065:. By
497:-like
455:-like
393:-like
362:Field
320:-like
294:Magma
262:Group
256:-like
254:Group
38:is a
3750:ISBN
3720:ISBN
3689:ISBN
3644:ISBN
3577:ISBN
3547:ISBN
3527:ISBN
3506:ISBN
3391:ISSN
3381:ISBN
3264:ISSN
3163:The
2921:The
2386:Let
2082:ring
2059:ring
1753:) â
1686:and
1576:), (
1562:), (
1425:The
1407:and
1399:Any
617:For
587:For
327:Ring
318:Ring
191:and
40:ring
34:, a
3730:Zbl
3681:doi
3662:Zbl
3636:doi
3609:doi
3569:doi
3498:doi
3407:Zbl
3373:doi
3256:doi
3179:).
3108:of
2961:.
2671:or
2128:is
2108:if
1912:If
1761:as
1729:to
1722:of
1483:Sym
1471:of
1316:of
1256:is
1138:If
1025:If
1016:.)
988:in
690:in
640:is
636:in
332:Rng
219:).
207:in
54:or
30:In
3793::
3778:,
3772:,
3748:,
3728:,
3699:MR
3697:.
3687:.
3660:,
3654:MR
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3642:,
3617:.
3605:46
3603:.
3597:.
3575:.
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3504:,
3492:,
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3389:.
3379:.
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3244:.
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2307:op
1986:A
1975:A
1956:,
1893:â
1652:,
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1634:â„
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238:.
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3705:.
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3157:H
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3141:R
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2810:(
2798:)
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2773:(
2770:=
2767:)
2764:f
2761:(
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2222:G
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2022:]
2019:G
2016:[
2013:R
1970:R
1966:R
1962:S
1958:Q
1954:Q
1950:S
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