2701:
with minor modifications. There is still a unique torsion part, with a torsionfree complement (unique up to isomorphism), but a torsionfree module over a
Dedekind domain is no longer necessarily free. Torsionfree modules over a Dedekind domain are determined (up to isomorphism) by rank and
835:
413:
2726:
which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary decomposition cannot hold for infinitely generated modules, even over the integers,
249:
1635:
2475:
2286:
1834:
1445:
1211:
526:
666:
2422:
2354:
457:. Since the ideals should be proper, these factors must not themselves be invertible (this avoids trivial factors in the sum), and the inclusion of the ideals means one has divisibility
970:
2228:
2710:(rank one free modules) is replaced by a direct sum into rank one projective modules: the individual summands are not uniquely determined, but the Steinitz class (of the sum) is.
2750:, the group of all sequences of integers under termwise addition. In general, the question of which infinitely generated torsion-free abelian groups are free depends on which
2527:
1937:
1538:
271:
1012:
2497:
1740:
580:
is actually a field, then all occurring ideals must be zero, and one obtains the decomposition of a finite dimensional vector space into a direct sum of one-dimensional
1312:
1364:
1244:
1045:
559:
2098:
1685:
1475:
1106:
1076:
896:
865:
647:
620:
443:
1940:
32:
1991:
1906:
2044:
1264:
2734:
Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free. For instance, consider the ring
2600:
For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ring generated by two elements. For the ring
2682:
into a direct sum of indecomposable modules is directly related (via the ideal class group) to the failure of the unique factorization of elements of
2718:
Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the number of factors may vary. There are
584:; the number of such factors is fixed, namely the dimension of the space, but there is a lot of freedom for choosing the subspaces themselves (if
177:
1582:
2561:
and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of
1318:, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a
2427:
2698:
2247:
148:
The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.
1760:
1376:
2873:
2854:
2807:
2778:
1119:
830:{\displaystyle R^{f}\oplus \bigoplus _{i}R/(d_{i})=R^{f}\oplus R/(d_{1})\oplus R/(d_{2})\oplus \cdots \oplus R/(d_{n-f})\;,}
653:
for the module. Explicitly, this means that any two modules sharing the same set of invariants are necessarily isomorphic.
460:
1548:
This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.
2186:, and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:
142:
2828:
2047:
2359:
2291:
2546:
is a more general result for finite groups (or modules over an arbitrary ring). In this generality, one obtains a
2134:
169:
920:
2192:
2163:
While the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between
129:
finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis of
2754:
exist. A consequence is that any structure theorem for infinitely generated modules depends on a choice of
2543:
2238:
137:.) By changing the choice of generating set, one can in fact describe the module as the quotient of some
2578:
2151:
650:
408:{\displaystyle M\cong \bigoplus _{i}R/(d_{i})=R/(d_{1})\oplus R/(d_{2})\oplus \cdots \oplus R/(d_{n}).}
2558:
2051:
1490:
110:
36:
2502:
1914:
1503:
2706:(which takes value in the ideal class group), and the decomposition into a direct sum of copies of
978:
2480:
1710:
2120:
1695:, then exists right inverse of the projection map (it suffices to lift each of the generators of
2878:
2747:
1327:
1276:
103:
43:
1882:
This includes the classification of finite-dimensional vector spaces as a special case, where
2802:, New York: Springer, pp. 218â226, Section IV.6: Modules over a Principal Ideal Domain,
2562:
2551:
2172:
2147:
2139:
2126:
2116:
1336:
1319:
1315:
1216:
1017:
531:
258:
2057:
2838:
2795:
2788:
1994:
1663:
1453:
1084:
1054:
874:
843:
625:
598:
421:
83:
54:. The result provides a simple framework to understand various canonical form results for
39:
1908:. Since fields have no non-trivial ideals, every finitely generated vector space is free.
8:
1961:
1885:
1649:
172:
59:
51:
2697:
the ideal class group is the only obstruction, and the structure theorem generalizes to
2547:
2529:
summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).
2234:
2020:
1249:
446:
2850:
2824:
2803:
2774:
2594:
2573:
The primary decomposition generalizes to finitely generated modules over commutative
2108:
1692:
1560:
1541:
528:. The free part is visible in the part of the decomposition corresponding to factors
1657:
2130:
2112:
450:
134:
24:
2834:
2816:
2784:
2694:
2574:
1704:
1323:
581:
118:
2751:
2703:
2168:
2101:
29:
structure theorem for finitely generated modules over a principal ideal domain
2867:
2746:-module which is not free. Another classical example of such a module is the
1747:
1494:
1048:
262:
55:
2176:
2143:
71:
2593:
submodules does not generalize as far, and the failure is measured by the
244:{\displaystyle (d_{1})\supseteq (d_{2})\supseteq \cdots \supseteq (d_{n})}
1653:
1630:{\displaystyle 0\rightarrow tM\rightarrow M\rightarrow M/tM\rightarrow 0}
20:
2823:(2 ed.), New York: W. H. Freeman and Company, pp. xviii+499,
2755:
1998:
91:
79:
1568:
47:
1116:. In a PID, nonzero primary ideals are powers of primes, and so
117:
might not exist. However such a module is still isomorphic to a
2616:
generated by 2 and 1 + ââ5 are indecomposable. While
2470:{\displaystyle {\begin{bmatrix}1&0\\1&1\end{bmatrix}}}
133:
to the generators of the module, and take the quotient by its
1493:
because a PID is
Noetherian, an even stronger condition than
46:(PID) can be uniquely decomposed in much the same way that
2849:, Graduate Texts in Mathematics No. 189, Springer-Verlag,
2281:{\displaystyle M\approx \mathbf {Z} \oplus \mathbf {Z} /2}
2175:
decomposition. This follows because there are non-trivial
1950:
be a linear operator on a finite-dimensional vector space
561:. Such factors, if any, occur at the end of the sequence.
1873:
is decomposed into cyclic modules of the indicated types.
1941:
fundamental theorem of finitely generated abelian groups
1829:{\displaystyle N_{p}=\{m\in tM\mid \exists i,mp^{i}=0\}}
1440:{\displaystyle R^{f}\oplus (\bigoplus _{i}R/(q_{i}))\;,}
1333:
As before, it is possible to write the free part (where
33:
fundamental theorem of finitely generated abelian groups
1206:{\displaystyle (q_{i})=(p_{i}^{r_{i}})=(p_{i})^{r_{i}}}
2436:
1652:
module, and such a module over a commutative PID is a
2505:
2483:
2430:
2362:
2294:
2250:
2195:
2179:
of these modules which do not preserve the summands.
2060:
2023:
1964:
1917:
1888:
1763:
1713:
1666:
1585:
1506:
1456:
1379:
1339:
1279:
1252:
1219:
1122:
1087:
1057:
1020:
981:
923:
877:
846:
669:
628:
601:
570:, the isomorphism giving the decomposition itself is
534:
521:{\displaystyle d_{1}\,|\,d_{2}\,|\,\cdots \,|\,d_{n}}
463:
424:
274:
180:
2758:
axioms and may be invalid under a different choice.
1489:Every finitely generated module over a PID is also
445:of the ideals are unique up to multiplication by a
2521:
2491:
2469:
2416:
2348:
2280:
2222:
2092:
2038:
2017:can be viewed as a finitely generated module over
1985:
1931:
1900:
1828:
1734:
1679:
1629:
1532:
1469:
1439:
1358:
1306:
1258:
1238:
1205:
1100:
1070:
1039:
1006:
964:
890:
859:
829:
641:
614:
553:
520:
437:
407:
243:
2713:
2699:finitely generated modules over a Dedekind domain
2424:is another basis, and the change of basis matrix
151:
2865:
2815:
2182:However, one has a canonical torsion submodule
564:While the direct sum is uniquely determined by
82:generating set, then one may extract from it a
2565:in which the summands are unique up to order.
2171:is not unique, and does not even preserve the
2050:, and the product of invariant factors is the
2417:{\displaystyle (1,{\bar {1}}),(0,{\bar {1}})}
2349:{\displaystyle (1,{\bar {0}}),(0,{\bar {1}})}
1326:, this can be seen as a manifestation of the
1078:are unique (up to multiplication by units).
2769:Dummit, David S.; Foote, Richard M. (2004),
1849:is a direct sum of cyclic modules, and that
1823:
1777:
2768:
2054:. Combined with a standard matrix form for
168:, there is a unique decreasing sequence of
2794:
2568:
1433:
965:{\displaystyle \bigoplus _{i}R/(q_{i})\;,}
958:
823:
2584:
1925:
1869:Putting the previous two steps together,
1540:(relations to generators), and put it in
1246:, the resulting indecomposable module is
898:'s in the original sequence which are 0.
507:
501:
497:
491:
480:
474:
2666:which give a different decomposition of
2644:summands give indecomposable submodules
901:
2589:By contrast, unique decomposition into
2223:{\displaystyle 0<\cdots <T<M.}
1862:for a finite number of distinct primes
1266:itself, and this is inside the part of
109:is no longer true, since a basis for a
90:of vectors, and the space is therefore
2866:
2674:. The failure of uniquely factorizing
656:Some prefer to write the free part of
622:elements, together with the number of
2009:, yields structure information about
145:, and this is the structure theorem.
1575:and this gives short exact sequence:
1500:Take a presentation, which is a map
156:For every finitely generated module
2844:
2046:. The last invariant factor is the
98:. The corresponding statement with
13:
2532:
1795:
14:
2890:
2773:(3rd ed.), New York: Wiley,
914:is isomorphic to one of the form
2577:, and this result is called the
2507:
2499:. However, it does preserve the
2485:
2266:
2258:
2150:(this latter only holds over an
1640:Where the map is a projection.
906:Every finitely generated module
2135:primary rational canonical form
1485:One proof proceeds as follows:
2714:Non-finitely generated modules
2522:{\displaystyle \mathbf {Z} /2}
2477:does not preserve the summand
2411:
2405:
2390:
2384:
2378:
2363:
2343:
2337:
2322:
2316:
2310:
2295:
2087:
2081:
2070:
2064:
2033:
2027:
1980:
1974:
1932:{\displaystyle R=\mathbb {Z} }
1877:
1691:. Since every free module is
1621:
1604:
1598:
1589:
1533:{\displaystyle R^{r}\to R^{g}}
1517:
1430:
1427:
1414:
1393:
1320:completely decomposable module
1301:
1288:
1187:
1173:
1167:
1142:
1136:
1123:
1034:
1021:
995:
982:
955:
942:
910:over a principal ideal domain
820:
801:
781:
768:
754:
741:
714:
701:
503:
493:
476:
399:
386:
366:
353:
339:
326:
312:
299:
238:
225:
213:
200:
194:
181:
162:over a principal ideal domain
152:Invariant factor decomposition
86:consisting of a finite number
1:
2847:Lectures on modules and rings
2761:
2686:into irreducible elements of
2158:
1840:, and it turns out that each
1007:{\displaystyle (q_{i})\neq R}
16:Statement in abstract algebra
2874:Theorems in abstract algebra
2492:{\displaystyle \mathbf {Z} }
1735:{\displaystyle M=tM\oplus F}
1707:(left split) M splits into:
1551:Another outline of a proof:
574:in general. For instance if
65:
7:
2597:, which vanishes for PIDs.
31:is a generalization of the
10:
2895:
2152:algebraically closed field
1836:. This is a submodule of
1660:, so it is isomorphic to:
651:complete set of invariants
2640:; thus the images of the
2537:
2052:characteristic polynomial
1480:
1366:) separately and express
1307:{\displaystyle R/(q_{i})}
141:by a particularly simple
111:finitely generated module
1648:is a finitely generated
1567:. Torsion module can be
35:and roughly states that
2121:rational canonical form
1687:for a positive integer
1359:{\displaystyle q_{i}=0}
1270:that is a free module.
1239:{\displaystyle q_{i}=0}
1040:{\displaystyle (q_{i})}
649:which are zero, form a
554:{\displaystyle d_{i}=0}
2585:Indecomposable modules
2579:LaskerâNoether theorem
2563:indecomposable modules
2523:
2493:
2471:
2418:
2350:
2282:
2224:
2100:, this yields various
2094:
2093:{\displaystyle K/p(T)}
2040:
1987:
1933:
1902:
1830:
1736:
1681:
1631:
1534:
1471:
1441:
1360:
1328:Lasker-Noether theorem
1308:
1260:
1240:
1207:
1102:
1072:
1041:
1008:
966:
892:
861:
831:
643:
616:
555:
522:
439:
409:
245:
104:principal ideal domain
44:principal ideal domain
2796:Hungerford, Thomas W.
2620:is not isomorphic to
2569:Primary decomposition
2559:KrullâSchmidt theorem
2544:JordanâHölder theorem
2524:
2494:
2472:
2419:
2351:
2283:
2239:JordanâHölder theorem
2225:
2148:Jordan canonical form
2140:primary decomposition
2127:primary decomposition
2117:Frobenius normal form
2095:
2041:
2001:with coefficients in
1988:
1934:
1903:
1831:
1757:we can then speak of
1737:
1682:
1680:{\displaystyle R^{n}}
1632:
1535:
1472:
1470:{\displaystyle q_{i}}
1442:
1361:
1309:
1261:
1241:
1208:
1103:
1101:{\displaystyle q_{i}}
1073:
1071:{\displaystyle q_{i}}
1042:
1009:
967:
902:Primary decomposition
893:
891:{\displaystyle d_{i}}
862:
860:{\displaystyle d_{i}}
832:
644:
642:{\displaystyle d_{i}}
617:
615:{\displaystyle d_{i}}
556:
523:
440:
438:{\displaystyle d_{i}}
410:
257:is isomorphic to the
246:
2503:
2481:
2428:
2360:
2292:
2248:
2193:
2058:
2021:
1962:
1915:
1886:
1761:
1711:
1664:
1583:
1504:
1454:
1377:
1337:
1277:
1250:
1217:
1120:
1085:
1055:
1018:
979:
921:
875:
844:
667:
626:
599:
532:
461:
422:
272:
178:
2845:Lam, T. Y. (1999),
2356:is one basis, then
1986:{\displaystyle R=K}
1901:{\displaystyle R=K}
1853:is a direct sum of
1166:
1110:elementary divisors
52:prime factorization
2748:BaerâSpecker group
2742:is a torsion-free
2738:of integers. Then
2612:and its submodule
2608:, both the module
2548:composition series
2519:
2489:
2467:
2461:
2414:
2346:
2278:
2235:composition series
2220:
2090:
2048:minimal polynomial
2036:
1983:
1929:
1898:
1826:
1732:
1677:
1627:
1571:as a submodule of
1530:
1491:finitely presented
1467:
1450:where the visible
1437:
1405:
1356:
1322:. Since PID's are
1304:
1256:
1236:
1203:
1145:
1098:
1068:
1037:
1004:
962:
933:
888:
857:
840:where the visible
827:
692:
639:
612:
551:
518:
435:
405:
290:
241:
37:finitely generated
23:, in the field of
2856:978-0-387-98428-5
2809:978-0-387-90518-1
2780:978-0-471-43334-7
2632:is isomorphic to
2595:ideal class group
2408:
2381:
2340:
2313:
2244:For instance, if
2109:invariant factors
2039:{\displaystyle K}
1693:projective module
1561:torsion submodule
1542:Smith normal form
1396:
1259:{\displaystyle R}
924:
871:is the number of
867:are nonzero, and
683:
451:invariant factors
449:, and are called
281:
102:generalized to a
2886:
2859:
2841:
2821:Basic algebra. I
2817:Jacobson, Nathan
2812:
2791:
2771:Abstract algebra
2693:However, over a
2658: <
2575:Noetherian rings
2550:, rather than a
2528:
2526:
2525:
2520:
2515:
2510:
2498:
2496:
2495:
2490:
2488:
2476:
2474:
2473:
2468:
2466:
2465:
2423:
2421:
2420:
2415:
2410:
2409:
2401:
2383:
2382:
2374:
2355:
2353:
2352:
2347:
2342:
2341:
2333:
2315:
2314:
2306:
2287:
2285:
2284:
2279:
2274:
2269:
2261:
2229:
2227:
2226:
2221:
2131:companion matrix
2113:companion matrix
2099:
2097:
2096:
2091:
2077:
2045:
2043:
2042:
2037:
1992:
1990:
1989:
1984:
1938:
1936:
1935:
1930:
1928:
1907:
1905:
1904:
1899:
1835:
1833:
1832:
1827:
1816:
1815:
1773:
1772:
1741:
1739:
1738:
1733:
1686:
1684:
1683:
1678:
1676:
1675:
1636:
1634:
1633:
1628:
1614:
1539:
1537:
1536:
1531:
1529:
1528:
1516:
1515:
1476:
1474:
1473:
1468:
1466:
1465:
1446:
1444:
1443:
1438:
1426:
1425:
1413:
1404:
1389:
1388:
1365:
1363:
1362:
1357:
1349:
1348:
1324:Noetherian rings
1313:
1311:
1310:
1305:
1300:
1299:
1287:
1265:
1263:
1262:
1257:
1245:
1243:
1242:
1237:
1229:
1228:
1212:
1210:
1209:
1204:
1202:
1201:
1200:
1199:
1185:
1184:
1165:
1164:
1163:
1153:
1135:
1134:
1107:
1105:
1104:
1099:
1097:
1096:
1077:
1075:
1074:
1069:
1067:
1066:
1046:
1044:
1043:
1038:
1033:
1032:
1013:
1011:
1010:
1005:
994:
993:
971:
969:
968:
963:
954:
953:
941:
932:
897:
895:
894:
889:
887:
886:
866:
864:
863:
858:
856:
855:
836:
834:
833:
828:
819:
818:
800:
780:
779:
767:
753:
752:
740:
729:
728:
713:
712:
700:
691:
679:
678:
648:
646:
645:
640:
638:
637:
621:
619:
618:
613:
611:
610:
591:
579:
569:
560:
558:
557:
552:
544:
543:
527:
525:
524:
519:
517:
516:
506:
496:
490:
489:
479:
473:
472:
444:
442:
441:
436:
434:
433:
414:
412:
411:
406:
398:
397:
385:
365:
364:
352:
338:
337:
325:
311:
310:
298:
289:
256:
250:
248:
247:
242:
237:
236:
212:
211:
193:
192:
167:
161:
25:abstract algebra
2894:
2893:
2889:
2888:
2887:
2885:
2884:
2883:
2864:
2863:
2862:
2857:
2831:
2810:
2781:
2764:
2752:large cardinals
2722:-submodules of
2716:
2695:Dedekind domain
2657:
2650:
2587:
2571:
2540:
2535:
2533:Generalizations
2511:
2506:
2504:
2501:
2500:
2484:
2482:
2479:
2478:
2460:
2459:
2454:
2448:
2447:
2442:
2432:
2431:
2429:
2426:
2425:
2400:
2399:
2373:
2372:
2361:
2358:
2357:
2332:
2331:
2305:
2304:
2293:
2290:
2289:
2270:
2265:
2257:
2249:
2246:
2245:
2194:
2191:
2190:
2161:
2102:canonical forms
2073:
2059:
2056:
2055:
2022:
2019:
2018:
1963:
1960:
1959:
1924:
1916:
1913:
1912:
1887:
1884:
1883:
1880:
1861:
1848:
1811:
1807:
1768:
1764:
1762:
1759:
1758:
1712:
1709:
1708:
1705:splitting lemma
1671:
1667:
1665:
1662:
1661:
1610:
1584:
1581:
1580:
1524:
1520:
1511:
1507:
1505:
1502:
1501:
1483:
1461:
1457:
1455:
1452:
1451:
1421:
1417:
1409:
1400:
1384:
1380:
1378:
1375:
1374:
1344:
1340:
1338:
1335:
1334:
1295:
1291:
1283:
1278:
1275:
1274:
1251:
1248:
1247:
1224:
1220:
1218:
1215:
1214:
1195:
1191:
1190:
1186:
1180:
1176:
1159:
1155:
1154:
1149:
1130:
1126:
1121:
1118:
1117:
1108:are called the
1092:
1088:
1086:
1083:
1082:
1062:
1058:
1056:
1053:
1052:
1028:
1024:
1019:
1016:
1015:
989:
985:
980:
977:
976:
949:
945:
937:
928:
922:
919:
918:
904:
882:
878:
876:
873:
872:
851:
847:
845:
842:
841:
808:
804:
796:
775:
771:
763:
748:
744:
736:
724:
720:
708:
704:
696:
687:
674:
670:
668:
665:
664:
633:
629:
627:
624:
623:
606:
602:
600:
597:
596:
585:
575:
565:
539:
535:
533:
530:
529:
512:
508:
502:
492:
485:
481:
475:
468:
464:
462:
459:
458:
429:
425:
423:
420:
419:
418:The generators
393:
389:
381:
360:
356:
348:
333:
329:
321:
306:
302:
294:
285:
273:
270:
269:
252:
232:
228:
207:
203:
188:
184:
179:
176:
175:
163:
157:
154:
121:of some module
68:
56:square matrices
17:
12:
11:
5:
2892:
2882:
2881:
2876:
2861:
2860:
2855:
2842:
2829:
2813:
2808:
2792:
2779:
2765:
2763:
2760:
2715:
2712:
2704:Steinitz class
2655:
2648:
2591:indecomposable
2586:
2583:
2570:
2567:
2539:
2536:
2534:
2531:
2518:
2514:
2509:
2487:
2464:
2458:
2455:
2453:
2450:
2449:
2446:
2443:
2441:
2438:
2437:
2435:
2413:
2407:
2404:
2398:
2395:
2392:
2389:
2386:
2380:
2377:
2371:
2368:
2365:
2345:
2339:
2336:
2330:
2327:
2324:
2321:
2318:
2312:
2309:
2303:
2300:
2297:
2277:
2273:
2268:
2264:
2260:
2256:
2253:
2231:
2230:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2169:canonical form
2160:
2157:
2156:
2155:
2137:
2124:
2089:
2086:
2083:
2080:
2076:
2072:
2069:
2066:
2063:
2035:
2032:
2029:
2026:
1982:
1979:
1976:
1973:
1970:
1967:
1927:
1923:
1920:
1897:
1894:
1891:
1879:
1876:
1875:
1874:
1867:
1857:
1844:
1825:
1822:
1819:
1814:
1810:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1771:
1767:
1731:
1728:
1725:
1722:
1719:
1716:
1674:
1670:
1638:
1637:
1626:
1623:
1620:
1617:
1613:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1577:
1576:
1546:
1545:
1527:
1523:
1519:
1514:
1510:
1498:
1482:
1479:
1464:
1460:
1448:
1447:
1436:
1432:
1429:
1424:
1420:
1416:
1412:
1408:
1403:
1399:
1395:
1392:
1387:
1383:
1355:
1352:
1347:
1343:
1316:indecomposable
1303:
1298:
1294:
1290:
1286:
1282:
1255:
1235:
1232:
1227:
1223:
1198:
1194:
1189:
1183:
1179:
1175:
1172:
1169:
1162:
1158:
1152:
1148:
1144:
1141:
1138:
1133:
1129:
1125:
1095:
1091:
1065:
1061:
1049:primary ideals
1036:
1031:
1027:
1023:
1003:
1000:
997:
992:
988:
984:
973:
972:
961:
957:
952:
948:
944:
940:
936:
931:
927:
903:
900:
885:
881:
854:
850:
838:
837:
826:
822:
817:
814:
811:
807:
803:
799:
795:
792:
789:
786:
783:
778:
774:
770:
766:
762:
759:
756:
751:
747:
743:
739:
735:
732:
727:
723:
719:
716:
711:
707:
703:
699:
695:
690:
686:
682:
677:
673:
636:
632:
609:
605:
550:
547:
542:
538:
515:
511:
505:
500:
495:
488:
484:
478:
471:
467:
432:
428:
416:
415:
404:
401:
396:
392:
388:
384:
380:
377:
374:
371:
368:
363:
359:
355:
351:
347:
344:
341:
336:
332:
328:
324:
320:
317:
314:
309:
305:
301:
297:
293:
288:
284:
280:
277:
263:cyclic modules
240:
235:
231:
227:
224:
221:
218:
215:
210:
206:
202:
199:
196:
191:
187:
183:
153:
150:
67:
64:
15:
9:
6:
4:
3:
2:
2891:
2880:
2879:Module theory
2877:
2875:
2872:
2871:
2869:
2858:
2852:
2848:
2843:
2840:
2836:
2832:
2830:0-7167-1480-9
2826:
2822:
2818:
2814:
2811:
2805:
2801:
2797:
2793:
2790:
2786:
2782:
2776:
2772:
2767:
2766:
2759:
2757:
2753:
2749:
2745:
2741:
2737:
2732:
2730:
2725:
2721:
2711:
2709:
2705:
2700:
2696:
2691:
2689:
2685:
2681:
2678: â
2677:
2673:
2670: â
2669:
2665:
2662: â
2661:
2654:
2647:
2643:
2639:
2636: â
2635:
2631:
2628: â
2627:
2623:
2619:
2615:
2611:
2607:
2604: =
2603:
2598:
2596:
2592:
2582:
2580:
2576:
2566:
2564:
2560:
2555:
2553:
2549:
2545:
2530:
2516:
2512:
2462:
2456:
2451:
2444:
2439:
2433:
2402:
2396:
2393:
2387:
2375:
2369:
2366:
2334:
2328:
2325:
2319:
2307:
2301:
2298:
2275:
2271:
2262:
2254:
2251:
2242:
2240:
2236:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2189:
2188:
2187:
2185:
2180:
2178:
2177:automorphisms
2174:
2170:
2166:
2153:
2149:
2145:
2144:Jordan blocks
2141:
2138:
2136:
2132:
2128:
2125:
2122:
2118:
2114:
2110:
2107:
2106:
2105:
2103:
2084:
2078:
2074:
2067:
2061:
2053:
2049:
2030:
2024:
2016:
2012:
2008:
2005:evaluated at
2004:
2000:
1996:
1977:
1971:
1968:
1965:
1957:
1953:
1949:
1944:
1942:
1921:
1918:
1909:
1895:
1892:
1889:
1872:
1868:
1865:
1860:
1856:
1852:
1847:
1843:
1839:
1820:
1817:
1812:
1808:
1804:
1801:
1798:
1792:
1789:
1786:
1783:
1780:
1774:
1769:
1765:
1756:
1752:
1749:
1748:prime element
1745:
1744:
1743:
1729:
1726:
1723:
1720:
1717:
1714:
1706:
1702:
1698:
1694:
1690:
1672:
1668:
1659:
1655:
1651:
1647:
1643:
1624:
1618:
1615:
1611:
1607:
1601:
1595:
1592:
1586:
1579:
1578:
1574:
1570:
1566:
1562:
1558:
1554:
1553:
1552:
1549:
1543:
1525:
1521:
1512:
1508:
1499:
1496:
1492:
1488:
1487:
1486:
1478:
1477:are nonzero.
1462:
1458:
1434:
1422:
1418:
1410:
1406:
1401:
1397:
1390:
1385:
1381:
1373:
1372:
1371:
1369:
1353:
1350:
1345:
1341:
1331:
1329:
1325:
1321:
1317:
1296:
1292:
1284:
1280:
1273:The summands
1271:
1269:
1253:
1233:
1230:
1225:
1221:
1196:
1192:
1181:
1177:
1170:
1160:
1156:
1150:
1146:
1139:
1131:
1127:
1115:
1111:
1093:
1089:
1081:The elements
1079:
1063:
1059:
1050:
1029:
1025:
1001:
998:
990:
986:
959:
950:
946:
938:
934:
929:
925:
917:
916:
915:
913:
909:
899:
883:
879:
870:
852:
848:
824:
815:
812:
809:
805:
797:
793:
790:
787:
784:
776:
772:
764:
760:
757:
749:
745:
737:
733:
730:
725:
721:
717:
709:
705:
697:
693:
688:
684:
680:
675:
671:
663:
662:
661:
659:
654:
652:
634:
630:
607:
603:
593:
589:
583:
578:
573:
568:
562:
548:
545:
540:
536:
513:
509:
498:
486:
482:
469:
465:
456:
452:
448:
430:
426:
402:
394:
390:
382:
378:
375:
372:
369:
361:
357:
349:
345:
342:
334:
330:
322:
318:
315:
307:
303:
295:
291:
286:
282:
278:
275:
268:
267:
266:
264:
260:
255:
233:
229:
222:
219:
216:
208:
204:
197:
189:
185:
174:
171:
166:
160:
149:
146:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
105:
101:
97:
93:
89:
85:
81:
77:
74:over a field
73:
63:
61:
57:
53:
49:
45:
41:
38:
34:
30:
26:
22:
2846:
2820:
2799:
2770:
2743:
2739:
2735:
2733:
2728:
2723:
2719:
2717:
2707:
2692:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2652:
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2601:
2599:
2590:
2588:
2572:
2556:
2541:
2243:
2232:
2183:
2181:
2164:
2162:
2014:
2010:
2006:
2002:
1955:
1951:
1947:
1945:
1910:
1881:
1870:
1863:
1858:
1854:
1850:
1845:
1841:
1837:
1754:
1750:
1700:
1696:
1688:
1650:torsion free
1645:
1641:
1639:
1572:
1564:
1556:
1550:
1547:
1484:
1449:
1367:
1332:
1272:
1267:
1113:
1109:
1080:
974:
911:
907:
905:
868:
839:
660:separately:
657:
655:
595:The nonzero
594:
587:
576:
571:
566:
563:
454:
417:
253:
164:
158:
155:
147:
138:
130:
126:
122:
114:
106:
99:
95:
87:
75:
72:vector space
69:
28:
18:
1999:polynomials
1939:yields the
1878:Corollaries
1654:free module
21:mathematics
2868:Categories
2762:References
2756:set theory
2552:direct sum
2173:direct sum
2159:Uniqueness
1656:of finite
1555:Denote by
572:not unique
251:such that
92:isomorphic
2406:¯
2379:¯
2338:¯
2311:¯
2263:⊕
2255:≈
2203:⋯
1958:. Taking
1796:∃
1793:∣
1784:∈
1727:⊕
1622:→
1605:→
1599:→
1590:→
1518:→
1495:coherence
1398:⨁
1391:⊕
999:≠
926:⨁
813:−
791:⊕
788:⋯
785:⊕
758:⊕
731:⊕
685:⨁
681:⊕
582:subspaces
499:⋯
376:⊕
373:⋯
370:⊕
343:⊕
283:⨁
279:≅
223:⊇
220:⋯
217:⊇
198:⊇
143:submodule
66:Statement
2819:(1985),
2798:(1980),
2233:Compare
2167:and its
1569:embedded
1014:and the
119:quotient
48:integers
2839:0780184
2800:Algebra
2789:2286236
2651:,
2146:yields
2133:yields
2115:yields
1995:algebra
1911:Taking
1213:. When
70:When a
50:have a
42:over a
40:modules
2853:
2837:
2827:
2806:
2787:
2777:
2538:Groups
2288:, and
2119:(aka,
1993:, the
1746:For a
1703:). By
1481:Proofs
1051:. The
975:where
590:> 1
173:ideals
170:proper
135:kernel
80:finite
78:has a
60:fields
27:, the
1954:over
1699:into
125:with
113:over
84:basis
58:over
2851:ISBN
2825:ISBN
2804:ISBN
2775:ISBN
2557:The
2542:The
2212:<
2206:<
2200:<
1946:Let
1697:M/tM
1658:rank
1559:the
1314:are
1047:are
586:dim
447:unit
2237:in
1997:of
1753:in
1563:of
1370:as
1112:of
592:).
453:of
261:of
259:sum
94:to
19:In
2870::
2835:MR
2833:,
2785:MR
2783:,
2731:.
2690:.
2624:,
2581:.
2554:.
2241:.
2142:+
2129:+
2111:+
2104::
2013:.
1943:.
1851:tM
1838:tM
1742:.
1646:tM
1557:tM
1330:.
265::
62:.
2744:Z
2740:Q
2736:Z
2729:Z
2724:Q
2720:Z
2708:R
2688:R
2684:R
2680:R
2676:R
2672:R
2668:R
2664:R
2660:R
2656:2
2653:L
2649:1
2646:L
2642:M
2638:M
2634:M
2630:R
2626:R
2622:M
2618:R
2614:M
2610:R
2606:Z
2602:R
2517:2
2513:/
2508:Z
2486:Z
2463:]
2457:1
2452:1
2445:0
2440:1
2434:[
2412:)
2403:1
2397:,
2394:0
2391:(
2388:,
2385:)
2376:1
2370:,
2367:1
2364:(
2344:)
2335:1
2329:,
2326:0
2323:(
2320:,
2317:)
2308:0
2302:,
2299:1
2296:(
2276:2
2272:/
2267:Z
2259:Z
2252:M
2218:.
2215:M
2209:T
2197:0
2184:T
2165:M
2154:)
2123:)
2088:)
2085:T
2082:(
2079:p
2075:/
2071:]
2068:T
2065:[
2062:K
2034:]
2031:T
2028:[
2025:K
2015:V
2011:T
2007:T
2003:K
1981:]
1978:T
1975:[
1972:K
1969:=
1966:R
1956:K
1952:V
1948:T
1926:Z
1922:=
1919:R
1896:K
1893:=
1890:R
1871:M
1866:.
1864:p
1859:p
1855:N
1846:p
1842:N
1824:}
1821:0
1818:=
1813:i
1809:p
1805:m
1802:,
1799:i
1790:M
1787:t
1781:m
1778:{
1775:=
1770:p
1766:N
1755:R
1751:p
1730:F
1724:M
1721:t
1718:=
1715:M
1701:M
1689:n
1673:n
1669:R
1644:/
1642:M
1625:0
1619:M
1616:t
1612:/
1608:M
1602:M
1596:M
1593:t
1587:0
1573:M
1565:M
1544:.
1526:g
1522:R
1513:r
1509:R
1497:.
1463:i
1459:q
1435:,
1431:)
1428:)
1423:i
1419:q
1415:(
1411:/
1407:R
1402:i
1394:(
1386:f
1382:R
1368:M
1354:0
1351:=
1346:i
1342:q
1302:)
1297:i
1293:q
1289:(
1285:/
1281:R
1268:M
1254:R
1234:0
1231:=
1226:i
1222:q
1197:i
1193:r
1188:)
1182:i
1178:p
1174:(
1171:=
1168:)
1161:i
1157:r
1151:i
1147:p
1143:(
1140:=
1137:)
1132:i
1128:q
1124:(
1114:M
1094:i
1090:q
1064:i
1060:q
1035:)
1030:i
1026:q
1022:(
1002:R
996:)
991:i
987:q
983:(
960:,
956:)
951:i
947:q
943:(
939:/
935:R
930:i
912:R
908:M
884:i
880:d
869:f
853:i
849:d
825:,
821:)
816:f
810:n
806:d
802:(
798:/
794:R
782:)
777:2
773:d
769:(
765:/
761:R
755:)
750:1
746:d
742:(
738:/
734:R
726:f
722:R
718:=
715:)
710:i
706:d
702:(
698:/
694:R
689:i
676:f
672:R
658:M
635:i
631:d
608:i
604:d
588:M
577:R
567:M
549:0
546:=
541:i
537:d
514:n
510:d
504:|
494:|
487:2
483:d
477:|
470:1
466:d
455:M
431:i
427:d
403:.
400:)
395:n
391:d
387:(
383:/
379:R
367:)
362:2
358:d
354:(
350:/
346:R
340:)
335:1
331:d
327:(
323:/
319:R
316:=
313:)
308:i
304:d
300:(
296:/
292:R
287:i
276:M
254:M
239:)
234:n
230:d
226:(
214:)
209:2
205:d
201:(
195:)
190:1
186:d
182:(
165:R
159:M
139:R
131:R
127:n
123:R
115:R
107:R
100:F
96:F
88:n
76:F
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