2775:
necessarily have finite uniform dimension. For example, an infinite direct product of nonzero rings is a finitely generated (cyclic!) module over itself, however it clearly contains an infinite direct sum of nonzero submodules. Finitely generated modules
701:
is a
Noetherian module (and indeed this property characterizes Noetherian rings): A module over a Noetherian ring is finitely generated if and only if it is a Noetherian module. This resembles, but is not exactly
1797:
is a direct sum of a torsion module and a projective module. A finitely generated projective module over a
Noetherian integral domain has constant rank and so the generic rank of a finitely generated module over
1501:
1657:
623:
1124:
1610:
2561:
2470:
2148:
2095:
2396:
2310:
2254:
1248:
2022:
1935:
1885:
1660:, the basic form of which says a finitely generated module over a PID is a direct sum of a torsion module and a free module. But it can also be shown directly as follows: let
414:
392:
1707:
3272:, Cambridge Studies in Advanced Mathematics, vol. 8, Translated from the Japanese by M. Reid (2 ed.), Cambridge: Cambridge University Press, pp. xiv+320,
332:
1769:
3006:
Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and a finitely related flat module is projective.
2501:
1416:
1639:
1276:
1730:
1374:
1335:
2771:. This is easily seen by applying the characterization using the finitely generated essential socle. Somewhat asymmetrically, finitely generated modules
682:
consisting of all those polynomials with zero constant term. Since every polynomial contains only finitely many terms whose coefficients are non-zero, the
360:
are needed to express the generators in any finite generating set, and these finitely many elements form a generating set. However, it may occur that
846:
is fundamental. Sometimes, the lemma allows one to prove finite dimensional vector spaces phenomena for finitely generated modules. For example, if
3047:
Although coherence seems like a more cumbersome condition than finitely generated or finitely presented, it is nicer than them since the
1425:
17:
2996:, coherent modules are finitely presented, and finitely presented modules are both finitely generated and finitely related. For a
721:. Conversely, if a finitely generated algebra is integral (over the coefficient ring), then it is finitely generated module. (See
714:
is
Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring.
1077:
3311:
3259:
3234:
1522:
477:
2781:
2510:
2419:
2104:
2051:
3277:
3200:
2345:
2259:
2203:
1504:
1160:
619:
117:
2575:
Both f.g. modules and f.cog. modules have interesting relationships to
Noetherian and Artinian modules, and the
1977:
532:. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module
1890:
1840:
2975:
283:
in this case. A finite generating set need not be a basis, since it need not be linearly independent over
3055:, while, in general, neither finitely generated nor finitely presented modules form an abelian category.
457:
352:
generates a module that is finitely generated, then there is a finite generating set that is included in
3074:
718:
703:
276:
46:
3193:
Commutative algebra. Chapters 1--7 Translated from the French. Reprint of the 1989 English translation
1512:
2594:) of a module. The following facts illustrate the duality between the two conditions. For a module
1153:
106:
397:
375:
2162:
From these conditions it is easy to see that being finitely generated is a property preserved by
1683:
417:
304:
3330:
3048:
1739:
1649:
796:
654:
of finitely generated modules need not be finitely generated. As an example, consider the ring
3003:, finitely generated, finitely presented, and coherent are equivalent conditions on a module.
2642:
2167:
924:
416:-module, and a generating set formed from prime numbers has at least two elements, while the
2177:. The following conditions are equivalent to a module being finitely cogenerated (f.cog.):
3287:
3218:
3181:
3069:
2479:
1786:
1394:
469:
445:
427:
39:
2988:
is a finitely generated module whose finitely generated submodules are finitely presented.
1615:
8:
3176:, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., pp. ix+128,
2967:
2672:
2587:
1653:
1253:
843:
647:
438:
292:
102:
1712:
2163:
1340:
1301:
874:
50:
923:-submodules. This is useful for weakening an assumption to the finite case (e.g., the
764:′′ is finitely presented (which is stronger than finitely generated; see below), then
98:
the concepts of finitely generated, finitely presented and coherent modules coincide.
3307:
3273:
3255:
3230:
3196:
2966:. Finitely presented modules can be characterized by an abstract property within the
2963:
2768:
2714:
2691:
1790:
694:
537:
529:
249:
3299:
3188:
3064:
3052:
2903:
is obtained by taking a free module and introducing finitely many relations within
2872:
2576:
1287:
1046:
935:
722:
577:
3101:
3283:
3214:
3177:
2997:
1779:
1002:
993:
itself. Because the ring product may be used to combine elements, more than just
916:
894:
717:
More generally, an algebra (e.g., ring) that is a finitely generated module is a
698:
525:
95:
1952:
3251:
3079:
1279:
901:
890:
741:
1658:
structure theorem for finitely generated modules over a principal ideal domain
3324:
3021:
2803:
697:
if every submodule is finitely generated. A finitely generated module over a
634:
550:
2473:
878:
484:
434:
369:
109:
1805:
2813:
2098:
928:
365:
31:
3009:
It is true also that the following conditions are equivalent for a ring
938:
can be found in commutative algebras. To say that a commutative algebra
756:′′ are finitely generated. There are some partial converses to this. If
637:
are precisely finite dimensional vector spaces (over the division ring).
3303:
3243:
859:
663:
43:
3125:
833:
667:
651:
811:
is finitely generated (resp. finitely presented) if and only if the
1778:
By the same argument as above, a finitely generated module over a
1496:{\displaystyle P_{M}(t)=\sum (\operatorname {dim} _{k}M_{n})t^{n}}
1376:-module. Then the rank of this free module is the generic rank of
650:
of a finitely generated module is finitely generated. In general,
612:
113:
3137:
3113:
2944:
is specified using finitely many generators (the images of the
1938:
608:
is
Noetherian, then every fractional ideal arises in this way.
678:-module (with {1} as generating set). Consider the submodule
528:
implies that every nonzero finitely generated module admits
600:
to be the product of the denominators of the generators of
291:
is finitely generated if and only if there is a surjective
3229:, Graduate Texts in Mathematics No. 189, Springer-Verlag,
2804:
Finitely presented, finitely related, and coherent modules
494:
is finitely generated if and only if any increasing chain
3298:, Lecture Notes in Mathematics, vol. 585, Springer,
2808:
Another formulation is this: a finitely generated module
1656:
if and only if it is free. This is a consequence of the
1146:
or equivalently the rank of a maximal free submodule of
1119:{\displaystyle \operatorname {dim} _{K}(M\otimes _{A}K)}
549:
If a module is generated by one element, it is called a
3213:, Boston, Mass.: Allyn and Bacon Inc., pp. x+180,
1806:
Equivalent definitions and finitely cogenerated modules
1664:
be a torsion-free finitely generated module over a PID
1066:
be a finitely generated module over an integral domain
838:
For finitely generated modules over a commutative ring
2763:. The same is true if "f.g." is replaced with "f.cog."
564:
its field of fractions. Then every finitely generated
364:
does not contain any finite generating set of minimal
2958:) and finitely many relations (the generators of ker(
2513:
2482:
2422:
2348:
2262:
2206:
2107:
2054:
1980:
1893:
1843:
1742:
1732:
is free since it is a submodule of a free module and
1715:
1686:
1618:
1605:{\displaystyle P_{M}(t)=F(t)\prod (1-t^{d_{i}})^{-1}}
1525:
1428:
1397:
1343:
1304:
1256:
1163:
1080:
400:
378:
307:
3195:, Elements of Mathematics, Berlin: Springer-Verlag,
1024:, then the following two statements are equivalent:
1138:. This number is the same as the number of maximal
934:An example of a link between finite generation and
2619:is Artinian if and only if every quotient module
2555:
2495:
2464:
2390:
2304:
2248:
2166:. The conditions are also convenient to define a
2142:
2089:
2016:
1929:
1879:
1763:
1724:
1701:
1633:
1604:
1495:
1410:
1368:
1329:
1270:
1242:
1118:
834:Finitely generated modules over a commutative ring
830:is finitely generated (resp. finitely presented).
408:
386:
326:
2556:{\displaystyle \phi :M\to \prod _{i\in F}N_{i}\,}
2465:{\displaystyle \phi :M\to \prod _{i\in I}N_{i}\,}
2143:{\displaystyle \phi :\bigoplus _{i\in F}R\to M\,}
2090:{\displaystyle \phi :\bigoplus _{i\in I}R\to M\,}
1793:; consequently, a finitely generated module over
1391:by finitely many homogeneous elements of degrees
3322:
2800:) is not a semisimple ring is a counterexample.
345:is a quotient of a free module of finite rank).
3171:
3131:
27:In algebra, module with a finite generating set
2767:Finitely cogenerated modules must have finite
2391:{\displaystyle \bigcap _{i\in I}N_{i}=\{0\}\,}
2305:{\displaystyle \bigcap _{i\in F}N_{i}=\{0\}\,}
2249:{\displaystyle \bigcap _{i\in I}N_{i}=\{0\}\,}
772:is Noetherian (resp. Artinian) if and only if
3098:For example, Matsumura uses this terminology.
2605:is Noetherian if and only if every submodule
1243:{\displaystyle (M/F)_{(0)}=M_{(0)}/F_{(0)}=0}
865:-endomorphism of a finitely generated module
633:Finitely generated (say left) modules over a
2384:
2378:
2298:
2292:
2242:
2236:
611:Finitely generated modules over the ring of
2736:has a minimal submodule, and any submodule
1810:The following conditions are equivalent to
1020:is a commutative algebra (with unity) over
112:, and a finitely generated module over the
2017:{\displaystyle \bigcup _{i\in I}N_{i}=M\,}
950:means that there exists a set of elements
3267:
3208:
3143:
3119:
2563:is a monomorphism for some finite subset
2552:
2461:
2387:
2301:
2245:
2150:is an epimorphism for some finite subset
2139:
2086:
2013:
1926:
1876:
622:. These are completely classified by the
402:
380:
3293:
3187:
3172:Atiyah, M. F.; Macdonald, I. G. (1969),
3155:
3107:
706:, which states that the polynomial ring
94:all of which are defined below. Over a
1930:{\displaystyle \sum _{i\in F}N_{i}=M\,}
1880:{\displaystyle \sum _{i\in I}N_{i}=M\,}
1789:) is torsion-free if and only if it is
1041:is both a finitely generated ring over
356:, since only finitely many elements in
14:
3323:
487:of its finitely generated submodules.
2840:Suppose now there is an epimorphism,
1387:is generated as algebra over a field
136:is finitely generated if there exist
49:. A finitely generated module over a
3242:
1802:is the rank of its projective part.
997:-linear combinations of elements of
780:′′ are Noetherian (resp. Artinian).
3224:
3174:Introduction to commutative algebra
2702:), it is f.g. if and only if f.cog.
1648:A finitely generated module over a
908:is also a surjective endomorphism.
693:In general, a module is said to be
478:dimension theorem for vector spaces
101:A finitely generated module over a
24:
3110:, Ch 1, §3, no. 6, Proposition 11.
977:such that the smallest subring of
25:
3342:
2899:), this basically expresses that
1814:being finitely generated (f.g.):
1142:-linearly independent vectors in
620:finitely generated abelian groups
580:: that is, there is some nonzero
1383:Now suppose the integral domain
507:stabilizes: i.e., there is some
118:finitely generated abelian group
3039:Every finitely presented right
1057:
768:′ is finitely generated. Also,
674:itself is a finitely generated
483:Any module is the union of the
3149:
3092:
2523:
2432:
2181:For any family of submodules {
2133:
2080:
1818:For any family of submodules {
1752:
1672:a maximal free submodule. Let
1628:
1622:
1590:
1563:
1557:
1551:
1542:
1536:
1480:
1454:
1445:
1439:
1363:
1347:
1324:
1308:
1229:
1223:
1208:
1202:
1189:
1183:
1179:
1164:
1113:
1094:
1001:are generated. For example, a
630:as the principal ideal domain.
318:
13:
1:
3227:Lectures on modules and rings
3085:
2812:is one for which there is an
2667:is f.cog. if and only if soc(
2323:For any chain of submodules {
1008:is finitely generated by {1,
904:: any injective endomorphism
641:
368:. For example the set of the
123:
3268:Matsumura, Hideyuki (1989),
3165:
2879:is finitely generated, then
2732:is f.cog. and nonzero, then
1070:with the field of fractions
925:characterization of flatness
444:, and the generating set is
409:{\displaystyle \mathbb {Z} }
387:{\displaystyle \mathbb {Z} }
80:finitely cogenerated modules
7:
3294:Springer, Tonny A. (1977),
3132:Atiyah & Macdonald 1969
3058:
2172:finitely cogenerated module
1702:{\displaystyle fM\subset F}
690:is not finitely generated.
560:be an integral domain with
543:
10:
3347:
3209:Kaplansky, Irving (1970),
3075:Countably generated module
3051:of coherent modules is an
2918:is finitely generated and
2709:is f.g. and nonzero, then
1422:is graded as well and let
760:is finitely generated and
719:finitely generated algebra
456:and is referred to as the
423:is also a generating set.
327:{\displaystyle R^{n}\to M}
84:finitely presented modules
3146:, p. 11, Theorem 17.
2974:: they are precisely the
2938:finitely presented module
1764:{\displaystyle f:M\to fM}
885:. This says simply that
748:is finitely generated if
503:of submodules with union
78:Related concepts include
36:finitely generated module
18:Finitely-generated module
2780:necessarily have finite
2717:and any quotient module
1771:is an isomorphism since
1515:, there is a polynomial
1154:Rank of an abelian group
1031:is a finitely generated
88:finitely related modules
3270:Commutative ring theory
2907:(the generators of ker(
2885:finitely related module
2633:is f.g. if and only if
2312:for some finite subset
2101:, then the restriction
1641:is the generic rank of
944:finitely generated ring
710:over a Noetherian ring
704:Hilbert's basis theorem
596:. Indeed, one can take
372:is a generating set of
2922:has finite rank (i.e.
2557:
2497:
2466:
2392:
2306:
2250:
2144:
2091:
2018:
1931:
1881:
1765:
1726:
1703:
1650:principal ideal domain
1635:
1606:
1497:
1412:
1370:
1331:
1290:, there is an element
1272:
1244:
1120:
919:of finitely generated
791:its subring such that
476:elements: this is the
426:In the case where the
410:
388:
328:
275:} is referred to as a
3036:is a coherent module.
2788:with unity such that
2643:superfluous submodule
2558:
2498:
2496:{\displaystyle N_{i}}
2467:
2393:
2307:
2251:
2145:
2092:
2019:
1932:
1882:
1785:(or more generally a
1766:
1727:
1704:
1636:
1607:
1513:Hilbert–Serre theorem
1498:
1413:
1411:{\displaystyle d_{i}}
1371:
1332:
1273:
1245:
1121:
1074:. Then the dimension
803:-module. Then a left
411:
389:
329:
73:module of finite type
56:may also be called a
2782:co-uniform dimension
2759:are f.g. then so is
2511:
2480:
2420:
2346:
2260:
2204:
2105:
2052:
1978:
1891:
1841:
1787:semi-hereditary ring
1740:
1713:
1684:
1634:{\displaystyle F(1)}
1616:
1523:
1426:
1395:
1341:
1302:
1254:
1161:
1078:
470:linearly independent
446:linearly independent
398:
376:
305:
3225:Lam, T. Y. (1999),
3043:module is coherent.
2673:essential submodule
1271:{\displaystyle M/F}
1014:but not as a module
472:generating set has
287:. What is true is:
3304:10.1007/BFb0095644
2553:
2541:
2493:
2462:
2450:
2388:
2364:
2302:
2278:
2246:
2222:
2164:Morita equivalence
2140:
2129:
2087:
2076:
2014:
1996:
1927:
1909:
1877:
1859:
1761:
1736:is a PID. But now
1725:{\displaystyle fM}
1722:
1699:
1631:
1602:
1493:
1408:
1366:
1327:
1286:is Noetherian, by
1268:
1240:
1116:
1047:integral extension
877:, and hence is an
618:coincide with the
530:maximal submodules
406:
384:
324:
163:such that for any
107:finite-dimensional
3313:978-3-540-08242-2
3261:978-0-201-55540-0
3236:978-0-387-98428-5
3211:Commutative rings
3189:Bourbaki, Nicolas
2978:in this category.
2964:free presentation
2914:If the kernel of
2891:is isomorphic to
2784:either: any ring
2769:uniform dimension
2715:maximal submodule
2698:) for any module
2692:semisimple module
2526:
2435:
2349:
2263:
2207:
2114:
2061:
1981:
1894:
1844:
1775:is torsion-free.
1369:{\displaystyle A}
1330:{\displaystyle M}
936:integral elements
744:of modules. Then
648:homomorphic image
624:structure theorem
538:Noetherian module
524:. This fact with
16:(Redirected from
3338:
3316:
3296:Invariant theory
3290:
3264:
3250:(3rd ed.),
3239:
3221:
3205:
3184:
3159:
3158:, Theorem 2.5.6.
3153:
3147:
3141:
3135:
3129:
3123:
3117:
3111:
3105:
3099:
3096:
3070:Artin–Rees lemma
3065:Integral element
3053:abelian category
2957:
2936:is said to be a
2931:
2863:and free module
2577:Jacobson radical
2562:
2560:
2559:
2554:
2551:
2550:
2540:
2502:
2500:
2499:
2494:
2492:
2491:
2471:
2469:
2468:
2463:
2460:
2459:
2449:
2397:
2395:
2394:
2389:
2374:
2373:
2363:
2311:
2309:
2308:
2303:
2288:
2287:
2277:
2255:
2253:
2252:
2247:
2232:
2231:
2221:
2149:
2147:
2146:
2141:
2128:
2096:
2094:
2093:
2088:
2075:
2036:
2023:
2021:
2020:
2015:
2006:
2005:
1995:
1937:for some finite
1936:
1934:
1933:
1928:
1919:
1918:
1908:
1886:
1884:
1883:
1878:
1869:
1868:
1858:
1770:
1768:
1767:
1762:
1731:
1729:
1728:
1723:
1708:
1706:
1705:
1700:
1640:
1638:
1637:
1632:
1611:
1609:
1608:
1603:
1601:
1600:
1588:
1587:
1586:
1585:
1535:
1534:
1502:
1500:
1499:
1494:
1492:
1491:
1479:
1478:
1466:
1465:
1438:
1437:
1417:
1415:
1414:
1409:
1407:
1406:
1375:
1373:
1372:
1367:
1362:
1361:
1336:
1334:
1333:
1328:
1323:
1322:
1288:generic freeness
1277:
1275:
1274:
1269:
1264:
1249:
1247:
1246:
1241:
1233:
1232:
1217:
1212:
1211:
1193:
1192:
1174:
1125:
1123:
1122:
1117:
1109:
1108:
1090:
1089:
972:
893:. Similarly, an
844:Nakayama's lemma
829:
723:integral element
592:is contained in
578:fractional ideal
422:
415:
413:
412:
407:
405:
393:
391:
390:
385:
383:
333:
331:
330:
325:
317:
316:
92:coherent modules
21:
3346:
3345:
3341:
3340:
3339:
3337:
3336:
3335:
3321:
3320:
3314:
3280:
3262:
3237:
3203:
3168:
3163:
3162:
3154:
3150:
3142:
3138:
3134:, Exercise 6.1.
3130:
3126:
3118:
3114:
3106:
3102:
3097:
3093:
3088:
3061:
3035:
2998:Noetherian ring
2983:coherent module
2976:compact objects
2949:
2923:
2806:
2546:
2542:
2530:
2512:
2509:
2508:
2487:
2483:
2481:
2478:
2477:
2455:
2451:
2439:
2421:
2418:
2417:
2405:= {0} for some
2403:
2369:
2365:
2353:
2347:
2344:
2343:
2328:
2283:
2279:
2267:
2261:
2258:
2257:
2227:
2223:
2211:
2205:
2202:
2201:
2186:
2118:
2106:
2103:
2102:
2065:
2053:
2050:
2049:
2030:
2025:
2001:
1997:
1985:
1979:
1976:
1975:
1960:
1955:of submodules {
1914:
1910:
1898:
1892:
1889:
1888:
1864:
1860:
1848:
1842:
1839:
1838:
1823:
1808:
1780:Dedekind domain
1741:
1738:
1737:
1714:
1711:
1710:
1685:
1682:
1681:
1617:
1614:
1613:
1593:
1589:
1581:
1577:
1576:
1572:
1530:
1526:
1524:
1521:
1520:
1505:Poincaré series
1487:
1483:
1474:
1470:
1461:
1457:
1433:
1429:
1427:
1424:
1423:
1402:
1398:
1396:
1393:
1392:
1354:
1350:
1342:
1339:
1338:
1315:
1311:
1303:
1300:
1299:
1260:
1255:
1252:
1251:
1222:
1218:
1213:
1201:
1197:
1182:
1178:
1170:
1162:
1159:
1158:
1104:
1100:
1085:
1081:
1079:
1076:
1075:
1060:
1003:polynomial ring
970:
961:
951:
917:inductive limit
895:Artinian module
836:
825:
816:
797:faithfully flat
699:Noetherian ring
644:
546:
519:
502:
468:means that any
420:
401:
399:
396:
395:
379:
377:
374:
373:
312:
308:
306:
303:
302:
274:
265:
258:
244:
236:
227:
221:
214:
208:
193:
184:
177:
158:
149:
142:
126:
96:Noetherian ring
28:
23:
22:
15:
12:
11:
5:
3344:
3334:
3333:
3319:
3318:
3312:
3291:
3278:
3265:
3260:
3252:Addison-Wesley
3240:
3235:
3222:
3206:
3201:
3185:
3167:
3164:
3161:
3160:
3148:
3144:Kaplansky 1970
3136:
3124:
3122:, Theorem 2.4.
3120:Matsumura 1989
3112:
3100:
3090:
3089:
3087:
3084:
3083:
3082:
3080:Finite algebra
3077:
3072:
3067:
3060:
3057:
3045:
3044:
3037:
3031:
3025:
2992:Over any ring
2990:
2989:
2979:
2962:)). See also:
2948:generators of
2912:
2857:
2856:
2838:
2837:
2805:
2802:
2765:
2764:
2745:
2726:
2703:
2684:
2662:
2628:
2614:
2573:
2572:
2549:
2545:
2539:
2536:
2533:
2529:
2525:
2522:
2519:
2516:
2507:module, then
2490:
2486:
2458:
2454:
2448:
2445:
2442:
2438:
2434:
2431:
2428:
2425:
2414:
2401:
2386:
2383:
2380:
2377:
2372:
2368:
2362:
2359:
2356:
2352:
2326:
2321:
2300:
2297:
2294:
2291:
2286:
2282:
2276:
2273:
2270:
2266:
2244:
2241:
2238:
2235:
2230:
2226:
2220:
2217:
2214:
2210:
2184:
2160:
2159:
2138:
2135:
2132:
2127:
2124:
2121:
2117:
2113:
2110:
2085:
2082:
2079:
2074:
2071:
2068:
2064:
2060:
2057:
2046:
2028:
2012:
2009:
2004:
2000:
1994:
1991:
1988:
1984:
1958:
1949:
1925:
1922:
1917:
1913:
1907:
1904:
1901:
1897:
1875:
1872:
1867:
1863:
1857:
1854:
1851:
1847:
1821:
1807:
1804:
1760:
1757:
1754:
1751:
1748:
1745:
1721:
1718:
1698:
1695:
1692:
1689:
1630:
1627:
1624:
1621:
1599:
1596:
1592:
1584:
1580:
1575:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1533:
1529:
1490:
1486:
1482:
1477:
1473:
1469:
1464:
1460:
1456:
1453:
1450:
1447:
1444:
1441:
1436:
1432:
1405:
1401:
1365:
1360:
1357:
1353:
1349:
1346:
1326:
1321:
1318:
1314:
1310:
1307:
1294:(depending on
1280:torsion module
1267:
1263:
1259:
1239:
1236:
1231:
1228:
1225:
1221:
1216:
1210:
1207:
1204:
1200:
1196:
1191:
1188:
1185:
1181:
1177:
1173:
1169:
1166:
1126:is called the
1115:
1112:
1107:
1103:
1099:
1096:
1093:
1088:
1084:
1059:
1056:
1055:
1054:
1036:
966:
959:
915:-module is an
891:Hopfian module
835:
832:
821:
787:be a ring and
742:exact sequence
668:countably many
643:
640:
639:
638:
631:
609:
554:
545:
542:
515:
498:
404:
382:
335:
334:
323:
320:
315:
311:
277:generating set
270:
263:
256:
240:
232:
225:
219:
212:
206:
189:
182:
175:
171:, there exist
154:
147:
140:
125:
122:
47:generating set
26:
9:
6:
4:
3:
2:
3343:
3332:
3331:Module theory
3329:
3328:
3326:
3315:
3309:
3305:
3301:
3297:
3292:
3289:
3285:
3281:
3279:0-521-36764-6
3275:
3271:
3266:
3263:
3257:
3253:
3249:
3245:
3241:
3238:
3232:
3228:
3223:
3220:
3216:
3212:
3207:
3204:
3202:3-540-64239-0
3198:
3194:
3190:
3186:
3183:
3179:
3175:
3170:
3169:
3157:
3156:Springer 1977
3152:
3145:
3140:
3133:
3128:
3121:
3116:
3109:
3108:Bourbaki 1998
3104:
3095:
3091:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3062:
3056:
3054:
3050:
3042:
3038:
3034:
3030:
3026:
3023:
3022:coherent ring
3019:
3016:
3015:
3014:
3012:
3007:
3004:
3002:
2999:
2995:
2987:
2984:
2980:
2977:
2973:
2971:
2965:
2961:
2956:
2952:
2947:
2943:
2939:
2935:
2930:
2926:
2921:
2917:
2913:
2910:
2906:
2902:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2869:
2868:
2866:
2862:
2859:for a module
2854:
2850:
2846:
2843:
2842:
2841:
2835:
2831:
2827:
2826:
2825:
2823:
2819:
2815:
2811:
2801:
2799:
2795:
2791:
2787:
2783:
2779:
2774:
2770:
2762:
2758:
2754:
2750:
2746:
2743:
2739:
2735:
2731:
2727:
2724:
2720:
2716:
2712:
2708:
2704:
2701:
2697:
2694:(such as soc(
2693:
2689:
2685:
2682:
2678:
2674:
2670:
2666:
2663:
2660:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2629:
2626:
2622:
2618:
2615:
2612:
2608:
2604:
2601:
2600:
2599:
2597:
2593:
2589:
2585:
2581:
2578:
2570:
2566:
2547:
2543:
2537:
2534:
2531:
2527:
2520:
2517:
2514:
2506:
2488:
2484:
2476:, where each
2475:
2456:
2452:
2446:
2443:
2440:
2436:
2429:
2426:
2423:
2415:
2412:
2408:
2404:
2381:
2375:
2370:
2366:
2360:
2357:
2354:
2350:
2341:
2337:
2333:
2329:
2322:
2319:
2315:
2295:
2289:
2284:
2280:
2274:
2271:
2268:
2264:
2239:
2233:
2228:
2224:
2218:
2215:
2212:
2208:
2199:
2195:
2191:
2187:
2180:
2179:
2178:
2176:
2173:
2169:
2165:
2157:
2153:
2136:
2130:
2125:
2122:
2119:
2115:
2111:
2108:
2100:
2083:
2077:
2072:
2069:
2066:
2062:
2058:
2055:
2047:
2044:
2040:
2035:
2031:
2010:
2007:
2002:
1998:
1992:
1989:
1986:
1982:
1973:
1969:
1965:
1961:
1954:
1950:
1947:
1943:
1940:
1923:
1920:
1915:
1911:
1905:
1902:
1899:
1895:
1873:
1870:
1865:
1861:
1855:
1852:
1849:
1845:
1836:
1832:
1828:
1824:
1817:
1816:
1815:
1813:
1803:
1801:
1796:
1792:
1788:
1784:
1781:
1776:
1774:
1758:
1755:
1749:
1746:
1743:
1735:
1719:
1716:
1696:
1693:
1690:
1687:
1679:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1646:
1644:
1625:
1619:
1597:
1594:
1582:
1578:
1573:
1569:
1566:
1560:
1554:
1548:
1545:
1539:
1531:
1527:
1518:
1514:
1510:
1506:
1488:
1484:
1475:
1471:
1467:
1462:
1458:
1451:
1448:
1442:
1434:
1430:
1421:
1403:
1399:
1390:
1386:
1381:
1379:
1358:
1355:
1351:
1344:
1319:
1316:
1312:
1305:
1297:
1293:
1289:
1285:
1281:
1265:
1261:
1257:
1237:
1234:
1226:
1219:
1214:
1205:
1198:
1194:
1186:
1175:
1171:
1167:
1156:
1155:
1149:
1145:
1141:
1137:
1133:
1129:
1110:
1105:
1101:
1097:
1091:
1086:
1082:
1073:
1069:
1065:
1052:
1048:
1044:
1040:
1037:
1034:
1030:
1027:
1026:
1025:
1023:
1019:
1015:
1012:} as a ring,
1011:
1007:
1004:
1000:
996:
992:
988:
984:
980:
976:
969:
965:
958:
954:
949:
945:
941:
937:
932:
930:
926:
922:
918:
914:
909:
907:
903:
899:
896:
892:
888:
884:
880:
876:
872:
868:
864:
861:
857:
853:
849:
845:
841:
831:
828:
824:
819:
814:
810:
806:
802:
798:
794:
790:
786:
781:
779:
775:
771:
767:
763:
759:
755:
751:
747:
743:
740:′′ → 0 be an
739:
735:
731:
726:
724:
720:
715:
713:
709:
705:
700:
696:
691:
689:
685:
681:
677:
673:
669:
665:
661:
658: =
657:
653:
649:
636:
635:division ring
632:
629:
625:
621:
617:
614:
610:
607:
603:
599:
595:
591:
587:
583:
579:
575:
571:
567:
563:
559:
555:
552:
551:cyclic module
548:
547:
541:
539:
535:
531:
527:
523:
518:
514:
510:
506:
501:
497:
493:
488:
486:
481:
479:
475:
471:
467:
463:
459:
455:
451:
447:
443:
440:
436:
432:
429:
424:
419:
371:
370:prime numbers
367:
363:
359:
355:
351:
346:
344:
340:
321:
313:
309:
301:
300:
299:
297:
295:
290:
286:
282:
278:
273:
269:
262:
255:
251:
246:
243:
239:
235:
231:
224:
218:
211:
205:
201:
197:
192:
188:
181:
174:
170:
166:
162:
157:
153:
146:
139:
135:
131:
121:
119:
115:
111:
108:
104:
99:
97:
93:
89:
85:
81:
76:
74:
70:
69:
63:
61:
55:
52:
48:
45:
41:
37:
33:
19:
3295:
3269:
3247:
3226:
3210:
3192:
3173:
3151:
3139:
3127:
3115:
3103:
3094:
3046:
3040:
3032:
3028:
3017:
3010:
3008:
3005:
3000:
2993:
2991:
2985:
2982:
2969:
2968:category of
2959:
2954:
2950:
2945:
2941:
2937:
2933:
2928:
2924:
2919:
2915:
2908:
2904:
2900:
2896:
2892:
2888:
2884:
2883:is called a
2880:
2876:
2864:
2860:
2858:
2852:
2848:
2844:
2839:
2833:
2829:
2821:
2817:
2809:
2807:
2797:
2793:
2789:
2785:
2777:
2772:
2766:
2760:
2756:
2752:
2748:
2741:
2737:
2733:
2729:
2722:
2718:
2710:
2706:
2699:
2695:
2687:
2680:
2676:
2668:
2664:
2658:
2654:
2650:
2646:
2638:
2634:
2630:
2624:
2620:
2616:
2610:
2606:
2602:
2595:
2591:
2583:
2579:
2574:
2568:
2564:
2504:
2474:monomorphism
2410:
2406:
2399:
2339:
2335:
2331:
2324:
2317:
2313:
2197:
2193:
2189:
2182:
2174:
2171:
2170:notion of a
2161:
2155:
2151:
2042:
2038:
2033:
2026:
1971:
1967:
1963:
1956:
1945:
1941:
1834:
1830:
1826:
1819:
1811:
1809:
1799:
1794:
1782:
1777:
1772:
1733:
1677:
1673:
1669:
1665:
1661:
1654:torsion-free
1647:
1642:
1516:
1508:
1419:
1388:
1384:
1382:
1377:
1298:) such that
1295:
1291:
1283:
1151:
1147:
1143:
1139:
1135:
1131:
1128:generic rank
1127:
1071:
1067:
1063:
1061:
1058:Generic rank
1050:
1042:
1038:
1032:
1028:
1021:
1017:
1013:
1009:
1005:
998:
994:
990:
986:
982:
978:
974:
967:
963:
956:
952:
947:
943:
939:
933:
920:
912:
910:
905:
897:
886:
882:
879:automorphism
870:
866:
862:
855:
851:
847:
839:
837:
826:
822:
817:
812:
808:
804:
800:
792:
788:
784:
782:
777:
773:
769:
765:
761:
757:
753:
749:
745:
737:
733:
729:
727:
716:
711:
707:
692:
687:
683:
679:
675:
671:
659:
655:
645:
627:
615:
605:
601:
597:
593:
589:
585:
581:
573:
569:
565:
561:
557:
536:is called a
533:
526:Zorn's lemma
521:
516:
512:
508:
504:
499:
495:
491:
489:
485:directed set
482:
473:
466:well-defined
465:
461:
454:well-defined
453:
449:
441:
435:vector space
430:
425:
361:
357:
353:
349:
347:
342:
338:
336:
293:
288:
284:
280:
271:
267:
260:
253:
247:
241:
237:
233:
229:
222:
216:
209:
203:
199:
195:
190:
186:
179:
172:
168:
164:
160:
155:
151:
144:
137:
133:
129:
127:
116:is simply a
110:vector space
105:is simply a
100:
91:
87:
83:
79:
77:
72:
67:
66:finite over
65:
59:
57:
53:
35:
29:
3244:Lang, Serge
3027:The module
3020:is a right
2814:epimorphism
2099:epimorphism
981:containing
929:Tor functor
725:for more.)
670:variables.
664:polynomials
568:-submodule
366:cardinality
296:-linear map
42:that has a
32:mathematics
3086:References
2679:, and soc(
1791:projective
1680:such that
1519:such that
1418:. Suppose
1337:is a free
860:surjective
695:Noetherian
652:submodules
642:Some facts
588:such that
511:such that
394:viewed as
124:Definition
3166:Textbooks
2887:. Since
2828:f :
2744:is f.cog.
2683:) is f.g.
2661:) is f.g.
2627:is f.cog.
2535:∈
2528:∏
2524:→
2515:ϕ
2444:∈
2437:∏
2433:→
2424:ϕ
2358:∈
2351:⋂
2272:∈
2265:⋂
2216:∈
2209:⋂
2134:→
2123:∈
2116:⨁
2109:ϕ
2081:→
2070:∈
2063:⨁
2056:ϕ
2037:for some
1990:∈
1983:⋃
1903:∈
1896:∑
1853:∈
1846:∑
1753:→
1694:⊂
1595:−
1570:−
1561:∏
1511:. By the
1468:
1452:∑
1356:−
1317:−
1157:). Since
1102:⊗
1092:
927:with the
902:coHopfian
875:injective
626:, taking
490:A module
458:dimension
418:singleton
348:If a set
337:for some
319:→
128:The left
3325:Category
3246:(1997),
3191:(1998),
3059:See also
3049:category
2972:-modules
2940:. Here,
2932:), then
2847: :
2824: :
2816:mapping
2671:) is an
1951:For any
873:is also
850: :
815:-module
807:-module
728:Let 0 →
686:-module
613:integers
544:Examples
228:+ ... +
132:-module
114:integers
3288:1011461
3248:Algebra
3219:0254021
3182:0242802
2871:If the
2725:is f.g.
2641:) is a
2613:is f.g.
2398:, then
2256:, then
2024:, then
1887:, then
1709:. Then
1612:. Then
1503:be the
1282:. When
1045:and an
1035:module.
962:, ...,
869:, then
662:of all
437:over a
266:, ...,
185:, ...,
150:, ...,
71:, or a
62:-module
58:finite
3310:
3286:
3276:
3258:
3233:
3217:
3199:
3180:
2873:kernel
2778:do not
2773:do not
2713:has a
2649:, and
2586:) and
2503:is an
2097:is an
1939:subset
1676:be in
799:right
646:Every
428:module
44:finite
40:module
2895:/ker(
2820:onto
2690:is a
2588:socle
2472:is a
2342:, if
2338:} in
2200:, if
2196:} in
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