1531:, which not only forces the right and left ideals to be linearly ordered, but also requires that there be only finitely many ideals in the chains of left and right ideals. Because of this historical precedent, some authors include the Artinian condition or finite composition length condition in their definitions of uniserial modules and rings.
1471:
This result, due to
Facchini, has been extended to infinite direct sums of uniserial modules by Příhoda in 2006. This extension involves the so-called quasismall uniserial modules. These modules were defined by Nguyen Viet Dung and Facchini, and their existence was proved by Puninski. The weak form
699:
This section will deal mainly with
Noetherian serial rings and their subclass, Artinian serial rings. In general, rings are first broken down into indecomposable rings. Once the structure of these rings are known, the decomposable rings are direct products of the indecomposable ones. Also, for
850:
to the upper triangular matrices over a division ring (note the similarity to the structure of
Noetherian serial rings in the preceding paragraph). A complete description of structure in the case of a circle quiver is beyond the scope of this article, but can be found in
555:
module over a serial ring is a direct sum of cyclic uniserial submodules (and hence is serial). If additionally the ring is assumed to be
Noetherian, the finitely presented and finitely generated modules coincide, and so all finitely generated modules are serial.
845:
Artinian serial ring structure is classified in cases depending on the quiver structure. It turns out that the quiver structure for a basic, indecomposable, Artinian serial ring is always a circle or a line. In the case of the line quiver, the ring is
780:
In 1975, Kirichenko and
Warfield independently and simultaneously published analyses of the structure of Noetherian, non-Artinian serial rings. The results were the same however the methods they used were very different from each other. The study of
1472:
of the Krull-Schmidt
Theorem holds not only for uniserial modules, but also for several other classes of modules (biuniform modules, cyclically presented modules over serial rings, kernels of morphisms between indecomposable
430:
Since rings with unity always have a maximal right ideal, a right uniserial ring is necessarily local. As noted before, a finitely generated right ideal can be generated by a single element, and so right uniserial rings are
563:. Being uniserial is preserved for quotients of rings and modules, but never for products. A direct summand of a serial module is not necessarily serial, as was proved by Puninski, but direct summands of
1629:
of the module. In a module with finite composition length, this has the effect of forcing the composition factors to be isomorphic, hence the "homogeneous" adjective. It turns out that a serial ring
1246:
1200:
764:
214:
494:
153:
113:
1619:
175:
if it is a right serial module over itself. Left uniserial and left serial rings are defined in a similar way, and are in general distinct from their right-sided counterparts.
1103:
1077:
988:
962:
793:
defined on serial rings were important tools. The core result states that a right
Noetherian, non-Artinian, basic, indecomposable serial ring can be described as a type of
654:
1458:
1383:
1048:
933:
1284:
243:
1304:
548:
is an
Artinian principal ideal ring. Nakayama showed that Artinian serial rings have this property on their modules, and that the converse is not true
294:
Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side (for example, uniserial, serial,
1542:
to refer to an
Artinian serial ring. Nakayama showed that all modules over such rings are serial. Artinian serial rings are sometimes called
1527:
was used to mean "Artinian principal ideal ring" even as recently as the 1970s. Köthe's paper also required a uniserial ring to have a unique
1812:
267:
A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen,
1523:(literally "one-series") during investigations of rings over which all modules are direct sums of cyclic submodules. For this reason,
1896:
1878:
1857:
1830:
1794:
1535:
302:) then it is assumed the condition holds on both the left and right. Unless otherwise specified, each ring in this article is a
288:
2014:
Příhoda, Pavel (2004), "Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension",
551:
The most general result, perhaps, on the modules of a serial ring is attributed to Drozd and
Warfield: it states that every
2078:
Puninski, Gennadi (2001b), "Some model theory over a nearly simple uniserial domain and decompositions of serial modules",
855:). To paraphrase the result as it appears there: A basic Artinian serial ring whose quiver is a circle is a homomorphic
2100:
Puninski, Gennadi (2001c), "Some model theory over an exceptional uniserial ring and decompositions of serial modules",
2036:
Příhoda, Pavel (2006), "A version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules",
1205:
1159:
719:
552:
184:
664:
504:
442:
571:
605:
2151:
1956:
Köthe, Gottfried (1935), "Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring. (German)",
118:
78:
54:
1564:
1109:
991:
362:
and thus is directly indecomposable. It is also easy to see that every finitely generated submodule of
335:
160:
2156:
1497:
860:
831:
798:
790:
527:
35:
1082:
1056:
967:
941:
856:
671:
597:
is a uniserial ring, and all Artinian principal ideal rings are serial rings, as is illustrated by
331:
628:
351:
2003:
1866:
1528:
1249:
39:
1553:
for a serial module with the additional property that for any two finitely generated submodules
1516:
559:
Being right serial is preserved under direct products of rings and modules, and preserved under
284:
1388:
1313:
1001:
886:
1892:
1874:
1853:
1826:
1808:
1790:
701:
587:
378:
343:
1269:
2129:
2109:
2087:
2065:
2045:
2023:
1993:
1985:
1965:
1943:
1921:
1626:
1543:
1473:
847:
805:
535:
531:
520:
280:
222:
24:
1289:
782:
681:
598:
303:
1493:
688:
594:
593:
Many examples of serial rings can be gleaned from the structure sections above. Every
542:
392:
359:
272:
2092:
2145:
2134:
1509:
609:
583:
560:
410:
307:
295:
276:
179:
2028:
936:
771:
678:
661:
657:
1948:
1784:
2120:
Warfield, Robert B. Jr. (1975), "Serial rings and finitely presented modules.",
1264:
1051:
794:
777:. This is why the results are phrased in terms of indecomposable, basic rings.
250:
50:
2113:
2070:
2049:
1998:
1512:
has "chain" as its namesake, but it is in general not related to chain rings.
786:
623:
396:
299:
1926:
1912:
Eisenbud, David; Griffith, Phillip (1971), "The structure of serial rings",
268:
46:
17:
1823:
Endomorphism rings and direct sum decompositions in some classes of modules
1633:
is a finite direct sum of homogeneously serial right ideals if and only if
1852:, Mathematical Surveys and Monographs, 65. American Mathematical Society,
1850:
Rings and things and a fine array of twentieth century associative algebra
1843:, Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag
171:
if it is uniserial as a right module over itself, and likewise called a
2007:
1969:
813:
260:
has been used differently from the above definition: for clarification
217:
1989:
1934:
Facchini, Alberto (1996), "Krull-Schmidt fails for serial modules",
1645:
matrix ring over a local serial ring. Such rings are also known as
261:
2056:
Puninski, G. T. (2002), "Artinian and Noetherian serial rings.",
530:(which are a special case of serial rings) are direct sums of
994:
notion can be defined: the modules are said to have the same
523:, which is a stronger condition than being a semilocal ring.
1766:
1670:
1976:
Nakayama, Tadasi (1941), "On Frobeniusean algebras. II.",
700:
semiperfect rings such as serial rings, the basic ring is
1500:. By the same token, uniserial modules have been called
1807:, Research Notes in Mathematics, vol. 44, Pitman,
16:"Chain ring" redirects here. For the bicycle part, see
716:
is known, the theory of Morita equivalence gives that
1567:
1391:
1316:
1292:
1272:
1208:
1162:
1085:
1059:
1004:
970:
944:
889:
722:
631:
567:
direct sums of uniserial modules are serial modules.
445:
245:. This ring is always serial, and is uniserial when
225:
187:
121:
81:
314:
Properties of uniserial and serial rings and modules
515:is a local, uniserial module. This indicates that
1694:
1613:
1452:
1377:
1298:
1278:
1240:
1194:
1097:
1071:
1042:
982:
956:
927:
859:of a "blow-up" of a basic, indecomposable, serial
758:
648:
488:
237:
208:
147:
107:
1496:, which are by definition commutative, uniserial
1484:Right uniserial rings can also be referred to as
417:is assumed to be Artinian or Noetherian, then End
57:. This means simply that for any two submodules
1546:, and they have a well-developed module theory.
534:. Later, Cohen and Kaplansky determined that a
1869:; Gubareni, Nadiya; Kirichenko, V. V. (2004),
1241:{\displaystyle V_{1}\oplus \dots \oplus V_{t}}
1195:{\displaystyle U_{1}\oplus \dots \oplus U_{n}}
838:on and above the diagonal, and entries from J(
1480:Notes on alternate, similar and related terms
1152:non-zero uniserial right modules over a ring
366:can be generated by a single element, and so
8:
1671:Hazewinkel, Gubareni & Kirichenko (2004)
759:{\displaystyle R\cong \mathrm {End} _{B}(P)}
291:, P. Příhoda, G. Puninski, and R. Warfield.
1665:References for each author can be found in
2102:Journal of the London Mathematical Society
1803:Chatters, A. W.; Hajarnavis, C.R. (1980),
1783:Frank W. Anderson; Kent R. Fuller (1992),
1767:Hazewinkel, Gubareni & Kirichenko 2004
526:Köthe showed that the modules of Artinian
209:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
2133:
2091:
2069:
2027:
1997:
1947:
1925:
1666:
1594:
1571:
1566:
1444:
1425:
1409:
1399:
1390:
1369:
1350:
1334:
1324:
1315:
1291:
1271:
1232:
1213:
1207:
1186:
1167:
1161:
1084:
1058:
1034:
1015:
1003:
969:
943:
919:
900:
888:
741:
730:
721:
633:
632:
630:
489:{\displaystyle R=\oplus _{i=1}^{n}e_{i}R}
477:
467:
456:
444:
224:
202:
201:
193:
189:
188:
186:
139:
126:
120:
99:
86:
80:
1742:
1730:
852:
1682:
1658:
1754:
1718:
1706:
812:) is nonzero. This matrix ring is a
586:is trivially uniserial, and likewise
7:
1871:Algebras, rings and modules. Vol. 1.
1519:and Keizo Asano introduced the term
318:It is immediate that in a uniserial
148:{\displaystyle N_{2}\subseteq N_{1}}
108:{\displaystyle N_{1}\subseteq N_{2}}
867:A decomposition uniqueness property
1614:{\displaystyle A/J(A)\cong B/J(B)}
1476:, couniformly presented modules.)
737:
734:
731:
574:holds in Noetherian serial rings.
541:has this property for its modules
178:An easy motivating example is the
14:
1647:primary decomposable serial rings
708:is a serial ring with basic ring
604:More exotic examples include the
439:necessarily factors in the form
1786:Rings and Categories of Modules
1492:. This latter term alludes to
1108:The following weak form of the
704:to the original ring. Thus if
2029:10.1016/j.jalgebra.2004.06.027
1891:, Kluwer Academic Publishers,
1873:, Kluwer Academic Publishers,
1789:, Springer, pp. 347–349,
1695:Chatters & Hajarnavis 1980
1608:
1602:
1585:
1579:
1441:
1435:
1429:
1418:
1406:
1392:
1366:
1360:
1354:
1343:
1331:
1317:
1098:{\displaystyle V\rightarrow U}
1089:
1072:{\displaystyle U\rightarrow V}
1063:
1031:
1024:
1012:
1005:
983:{\displaystyle V\rightarrow U}
974:
957:{\displaystyle U\rightarrow V}
948:
916:
909:
897:
890:
753:
747:
643:
637:
163:of uniserial modules. A ring
1:
2093:10.1016/s0022-4049(00)00140-7
1949:10.1090/s0002-9947-96-01740-0
797:over a Noetherian, uniserial
2135:10.1016/0021-8693(75)90074-5
649:{\displaystyle \mathbb {F} }
1887:Puninski, Gennadi (2001a),
1805:Rings with chain conditions
1551:homogeneously serial module
1534:Expanding on Köthe's work,
770:is some finitely generated
435:rings. A right serial ring
2173:
1821:Facchini, Alberto (1998),
1540:generalized uniserial ring
879:are said to have the same
570:It has been verified that
287:, H. Kuppisch, I. Murase,
15:
2114:10.1112/s0024610701002344
2050:10.1080/00927870500455049
1841:Algebra. II. Ring theory.
1453:{\displaystyle _{e}=_{e}}
1378:{\displaystyle _{m}=_{m}}
1043:{\displaystyle _{e}=_{e}}
928:{\displaystyle _{m}=_{m}}
606:upper triangular matrices
395:which is very close to a
330:and 0 are simultaneously
2058:J. Math. Sci. (New York)
1637:is isomorphic to a full
1255:-modules if and only if
326:, all submodules except
155:. A module is called a
2071:10.1023/A:1014906008243
1936:Trans. Amer. Math. Soc.
1927:10.2140/pjm.1971.36.109
1549:Warfield used the term
1279:{\displaystyle \sigma }
1156:. Then the direct sums
712:, and the structure of
1615:
1454:
1379:
1300:
1280:
1242:
1196:
1099:
1073:
1044:
984:
958:
929:
760:
650:
490:
239:
238:{\displaystyle n>1}
210:
149:
109:
2080:J. Pure Appl. Algebra
1978:Annals of Mathematics
1825:, Birkhäuser Verlag,
1616:
1504:, and serial modules
1490:right valuation rings
1455:
1380:
1301:
1299:{\displaystyle \tau }
1281:
1243:
1197:
1110:Krull-Schmidt theorem
1100:
1074:
1050:, if there exists an
1045:
985:
959:
930:
761:
651:
572:Jacobson's conjecture
528:principal ideal rings
491:
399:in the sense that End
377:It is known that the
306:, and each module is
240:
211:
150:
110:
1848:Faith, Carl (1999),
1839:Faith, Carl (1976),
1565:
1389:
1314:
1290:
1270:
1263:and there exist two
1206:
1160:
1083:
1057:
1002:
968:
942:
935:, if there exists a
887:
861:quasi-Frobenius ring
720:
629:
590:are serial modules.
443:
411:maximal right ideals
279:, Phillip Griffith,
223:
185:
169:right uniserial ring
119:
79:
1867:Hazewinkel, Michiel
1508:. The notion of a
1079:and an epimorphism
964:and a monomorphism
472:
1999:10338.dmlcz/140501
1970:10.1007/bf01201343
1611:
1529:composition series
1450:
1375:
1296:
1276:
1238:
1192:
1095:
1069:
1040:
980:
954:
925:
834:with entries from
830:, and consists of
756:
646:
588:semisimple modules
561:quotients of rings
553:finitely presented
505:idempotent element
486:
452:
358:is also clearly a
283:, V.V Kirichenko,
235:
206:
145:
105:
1980:, Second Series,
1942:(11): 4561–4575,
1814:978-0-273-08446-4
1544:Nakayama algebras
1506:semichain modules
1486:right chain rings
1474:injective modules
702:Morita equivalent
532:cyclic submodules
427:is a local ring.
379:endomorphism ring
344:maximal submodule
173:right serial ring
2164:
2138:
2137:
2116:
2096:
2095:
2074:
2073:
2052:
2044:(4): 1479–1487,
2032:
2031:
2010:
2001:
1972:
1952:
1951:
1930:
1929:
1914:Pacific J. Math.
1901:
1883:
1862:
1844:
1835:
1817:
1799:
1770:
1764:
1758:
1752:
1746:
1740:
1734:
1728:
1722:
1716:
1710:
1704:
1698:
1692:
1686:
1680:
1674:
1667:Puninski (2001a)
1663:
1627:Jacobson radical
1625:(−) denotes the
1620:
1618:
1617:
1612:
1598:
1575:
1536:Tadashi Nakayama
1459:
1457:
1456:
1451:
1449:
1448:
1439:
1438:
1414:
1413:
1404:
1403:
1384:
1382:
1381:
1376:
1374:
1373:
1364:
1363:
1339:
1338:
1329:
1328:
1305:
1303:
1302:
1297:
1285:
1283:
1282:
1277:
1247:
1245:
1244:
1239:
1237:
1236:
1218:
1217:
1201:
1199:
1198:
1193:
1191:
1190:
1172:
1171:
1104:
1102:
1101:
1096:
1078:
1076:
1075:
1070:
1049:
1047:
1046:
1041:
1039:
1038:
1020:
1019:
989:
987:
986:
981:
963:
961:
960:
955:
934:
932:
931:
926:
924:
923:
905:
904:
806:Jacobson radical
765:
763:
762:
757:
746:
745:
740:
655:
653:
652:
647:
636:
599:semisimple rings
536:commutative ring
521:semiperfect ring
495:
493:
492:
487:
482:
481:
471:
466:
409:has at most two
244:
242:
241:
236:
215:
213:
212:
207:
205:
197:
192:
154:
152:
151:
146:
144:
143:
131:
130:
114:
112:
111:
106:
104:
103:
91:
90:
29:uniserial module
25:abstract algebra
2172:
2171:
2167:
2166:
2165:
2163:
2162:
2161:
2142:
2141:
2119:
2099:
2077:
2055:
2035:
2013:
1990:10.2307/1968984
1975:
1955:
1933:
1911:
1908:
1906:Primary Sources
1899:
1886:
1881:
1865:
1860:
1847:
1838:
1833:
1820:
1815:
1802:
1797:
1782:
1779:
1774:
1773:
1765:
1761:
1753:
1749:
1741:
1737:
1729:
1725:
1717:
1713:
1705:
1701:
1693:
1689:
1681:
1677:
1664:
1660:
1655:
1641: ×
1563:
1562:
1517:Gottfried Köthe
1494:valuation rings
1482:
1440:
1421:
1405:
1395:
1387:
1386:
1365:
1346:
1330:
1320:
1312:
1311:
1288:
1287:
1268:
1267:
1228:
1209:
1204:
1203:
1182:
1163:
1158:
1157:
1143:
1134:
1127:
1118:
1081:
1080:
1055:
1054:
1030:
1011:
1000:
999:
966:
965:
940:
939:
915:
896:
885:
884:
869:
821:
729:
718:
717:
697:
627:
626:
617:
580:
512:
501:
473:
441:
440:
422:
404:
386:
316:
304:ring with unity
275:, A. Facchini,
221:
220:
183:
182:
135:
122:
117:
116:
95:
82:
77:
76:
70:
63:
51:totally ordered
21:
12:
11:
5:
2170:
2168:
2160:
2159:
2154:
2144:
2143:
2140:
2139:
2128:(2): 187–222,
2117:
2108:(2): 311–326,
2097:
2086:(3): 319–337,
2075:
2053:
2033:
2011:
1973:
1953:
1931:
1907:
1904:
1903:
1902:
1897:
1884:
1879:
1863:
1858:
1845:
1836:
1831:
1818:
1813:
1800:
1795:
1778:
1775:
1772:
1771:
1759:
1747:
1735:
1723:
1711:
1699:
1687:
1675:
1657:
1656:
1654:
1651:
1610:
1607:
1604:
1601:
1597:
1593:
1590:
1587:
1584:
1581:
1578:
1574:
1570:
1538:used the term
1515:In the 1930s,
1481:
1478:
1447:
1443:
1437:
1434:
1431:
1428:
1424:
1420:
1417:
1412:
1408:
1402:
1398:
1394:
1372:
1368:
1362:
1359:
1356:
1353:
1349:
1345:
1342:
1337:
1333:
1327:
1323:
1319:
1306:of 1, 2, ...,
1295:
1275:
1235:
1231:
1227:
1224:
1221:
1216:
1212:
1189:
1185:
1181:
1178:
1175:
1170:
1166:
1139:
1132:
1123:
1116:
1094:
1091:
1088:
1068:
1065:
1062:
1037:
1033:
1029:
1026:
1023:
1018:
1014:
1010:
1007:
979:
976:
973:
953:
950:
947:
922:
918:
914:
911:
908:
903:
899:
895:
892:
881:monogeny class
868:
865:
817:
785:, Noetherian,
755:
752:
749:
744:
739:
736:
733:
728:
725:
696:
693:
689:Sylow subgroup
665:characteristic
645:
642:
639:
635:
613:
595:valuation ring
579:
576:
543:if and only if
510:
499:
485:
480:
476:
470:
465:
462:
459:
455:
451:
448:
418:
400:
393:semilocal ring
382:
360:uniform module
315:
312:
234:
231:
228:
204:
200:
196:
191:
142:
138:
134:
129:
125:
102:
98:
94:
89:
85:
68:
61:
13:
10:
9:
6:
4:
3:
2:
2169:
2158:
2155:
2153:
2152:Module theory
2150:
2149:
2147:
2136:
2131:
2127:
2123:
2118:
2115:
2111:
2107:
2103:
2098:
2094:
2089:
2085:
2081:
2076:
2072:
2067:
2064:: 2330–2347,
2063:
2059:
2054:
2051:
2047:
2043:
2039:
2038:Comm. Algebra
2034:
2030:
2025:
2021:
2017:
2012:
2009:
2005:
2000:
1995:
1991:
1987:
1983:
1979:
1974:
1971:
1967:
1963:
1959:
1954:
1950:
1945:
1941:
1937:
1932:
1928:
1923:
1919:
1915:
1910:
1909:
1905:
1900:
1898:0-7923-7187-9
1894:
1890:
1885:
1882:
1880:1-4020-2690-0
1876:
1872:
1868:
1864:
1861:
1859:0-8218-0993-8
1855:
1851:
1846:
1842:
1837:
1834:
1832:3-7643-5908-0
1828:
1824:
1819:
1816:
1810:
1806:
1801:
1798:
1796:0-387-97845-3
1792:
1788:
1787:
1781:
1780:
1776:
1768:
1763:
1760:
1756:
1751:
1748:
1744:
1743:Warfield 1975
1739:
1736:
1732:
1731:Nakayama 1941
1727:
1724:
1720:
1715:
1712:
1708:
1703:
1700:
1696:
1691:
1688:
1684:
1679:
1676:
1672:
1668:
1662:
1659:
1652:
1650:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1605:
1599:
1595:
1591:
1588:
1582:
1576:
1572:
1568:
1560:
1556:
1552:
1547:
1545:
1541:
1537:
1532:
1530:
1526:
1522:
1518:
1513:
1511:
1510:catenary ring
1507:
1503:
1502:chain modules
1499:
1495:
1491:
1487:
1479:
1477:
1475:
1469:
1467:
1464:= 1, 2, ...,
1463:
1445:
1432:
1426:
1422:
1415:
1410:
1400:
1396:
1370:
1357:
1351:
1347:
1340:
1335:
1325:
1321:
1309:
1293:
1273:
1266:
1262:
1258:
1254:
1251:
1233:
1229:
1225:
1222:
1219:
1214:
1210:
1187:
1183:
1179:
1176:
1173:
1168:
1164:
1155:
1151:
1147:
1142:
1138:
1131:
1126:
1122:
1115:
1111:
1106:
1092:
1086:
1066:
1060:
1053:
1035:
1027:
1021:
1016:
1008:
997:
996:epigeny class
993:
977:
971:
951:
945:
938:
920:
912:
906:
901:
893:
882:
878:
874:
866:
864:
862:
858:
854:
853:Puninski 2002
849:
843:
841:
837:
833:
829:
825:
820:
815:
811:
807:
803:
800:
796:
792:
789:, as well as
788:
784:
778:
776:
773:
769:
750:
742:
726:
723:
715:
711:
707:
703:
694:
692:
690:
686:
683:
680:
676:
673:
669:
666:
663:
659:
640:
625:
621:
616:
611:
610:division ring
607:
602:
600:
596:
591:
589:
585:
584:simple module
577:
575:
573:
568:
566:
562:
557:
554:
549:
547:
544:
540:
537:
533:
529:
524:
522:
518:
514:
506:
502:
483:
478:
474:
468:
463:
460:
457:
453:
449:
446:
438:
434:
428:
426:
421:
416:
412:
408:
403:
398:
394:
390:
385:
380:
375:
373:
372:Bézout module
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
329:
325:
321:
313:
311:
309:
305:
301:
297:
292:
290:
286:
282:
278:
274:
271:, Yu. Drozd,
270:
265:
263:
259:
254:
252:
248:
232:
229:
226:
219:
198:
194:
181:
180:quotient ring
176:
174:
170:
166:
162:
158:
157:serial module
140:
136:
132:
127:
123:
100:
96:
92:
87:
83:
74:
67:
60:
56:
52:
48:
44:
41:
37:
33:
30:
26:
19:
2125:
2121:
2105:
2101:
2083:
2079:
2061:
2057:
2041:
2037:
2019:
2015:
1981:
1977:
1961:
1957:
1939:
1935:
1917:
1913:
1889:Serial rings
1888:
1870:
1849:
1840:
1822:
1804:
1785:
1762:
1750:
1738:
1726:
1714:
1702:
1690:
1683:Příhoda 2004
1678:
1661:
1646:
1642:
1638:
1634:
1630:
1622:
1558:
1554:
1550:
1548:
1539:
1533:
1524:
1520:
1514:
1505:
1501:
1489:
1485:
1483:
1470:
1465:
1461:
1307:
1265:permutations
1260:
1256:
1252:
1153:
1149:
1145:
1140:
1136:
1129:
1124:
1120:
1113:
1107:
995:
937:monomorphism
880:
876:
872:
871:Two modules
870:
844:
839:
835:
827:
823:
818:
809:
801:
779:
774:
772:progenerator
767:
713:
709:
705:
698:
684:
674:
667:
658:finite field
619:
614:
603:
592:
581:
569:
564:
558:
550:
545:
538:
525:
516:
508:
497:
436:
433:right Bézout
432:
429:
424:
419:
414:
406:
401:
388:
383:
376:
371:
367:
363:
355:
352:local module
347:
339:
327:
323:
319:
317:
293:
281:I. Kaplansky
266:
257:
255:
246:
177:
172:
168:
167:is called a
164:
156:
72:
65:
58:
42:
31:
28:
22:
2157:Ring theory
2022:: 332–341,
1984:(1): 1–21,
1920:: 109–121,
1112:holds. Let
1052:epimorphism
826:) for some
795:matrix ring
787:prime rings
496:where each
336:superfluous
289:T. Nakayama
277:A.W. Goldie
273:D. Eisenbud
251:prime power
159:if it is a
2146:Categories
2122:J. Algebra
2016:J. Algebra
1755:Faith 1976
1719:Köthe 1935
1707:Faith 1999
1460:for every
1310:such that
1250:isomorphic
998:, denoted
883:, denoted
848:isomorphic
783:hereditary
624:group ring
622:, and the
519:is also a
397:local ring
300:Noetherian
161:direct sum
47:submodules
1964:: 31–44,
1777:Textbooks
1589:≅
1525:uniserial
1521:Einreihig
1427:τ
1352:σ
1294:τ
1274:σ
1226:⊕
1223:⋯
1220:⊕
1180:⊕
1177:⋯
1174:⊕
1090:→
1064:→
975:→
949:→
842:) below.
727:≅
695:Structure
677:having a
656:for some
454:⊕
332:essential
269:P.M. Cohn
262:see below
258:uniserial
256:The term
133:⊆
93:⊆
75:, either
55:inclusion
18:Chainring
1958:Math. Z.
832:matrices
804:, whose
578:Examples
322:-module
296:Artinian
285:G. Köthe
216:for any
45:, whose
2008:1968984
1498:domains
1135:, ...,
1119:, ...,
814:subring
791:quivers
618:
608:over a
423:
405:
387:
346:, then
218:integer
38:over a
2006:
1895:
1877:
1856:
1829:
1811:
1793:
1621:where
990:. The
799:domain
766:where
682:normal
679:cyclic
565:finite
503:is an
413:. If
342:has a
338:. If
308:unital
36:module
2004:JSTOR
1653:Notes
857:image
672:group
662:prime
391:is a
370:is a
350:is a
249:is a
34:is a
1893:ISBN
1875:ISBN
1854:ISBN
1827:ISBN
1809:ISBN
1791:ISBN
1669:and
1557:and
1385:and
1286:and
1248:are
1202:and
992:dual
875:and
816:of M
670:and
582:Any
507:and
334:and
230:>
64:and
49:are
40:ring
27:, a
2130:doi
2110:doi
2088:doi
2084:163
2066:doi
2062:110
2046:doi
2024:doi
2020:281
1994:hdl
1986:doi
1966:doi
1944:doi
1940:348
1922:doi
1488:or
1144:be
660:of
381:End
354:.
115:or
71:of
53:by
23:In
2148::
2126:37
2124:,
2106:64
2104:,
2082:,
2060:,
2042:34
2040:,
2018:,
2002:,
1992:,
1982:42
1962:39
1960:,
1938:,
1918:36
1916:,
1649:.
1561:,
1468:.
1259:=
1148:+
1128:,
1105:.
863:.
808:J(
691:.
601:.
374:.
310:.
298:,
264:.
253:.
2132::
2112::
2090::
2068::
2048::
2026::
1996::
1988::
1968::
1946::
1924::
1769:.
1757:.
1745:.
1733:.
1721:.
1709:.
1697:.
1685:.
1673:.
1643:n
1639:n
1635:R
1631:R
1623:J
1609:)
1606:B
1603:(
1600:J
1596:/
1592:B
1586:)
1583:A
1580:(
1577:J
1573:/
1569:A
1559:B
1555:A
1466:n
1462:i
1446:e
1442:]
1436:)
1433:i
1430:(
1423:V
1419:[
1416:=
1411:e
1407:]
1401:i
1397:U
1393:[
1371:m
1367:]
1361:)
1358:i
1355:(
1348:V
1344:[
1341:=
1336:m
1332:]
1326:i
1322:U
1318:[
1308:n
1261:t
1257:n
1253:R
1234:t
1230:V
1215:1
1211:V
1188:n
1184:U
1169:1
1165:U
1154:R
1150:t
1146:n
1141:t
1137:V
1133:1
1130:V
1125:n
1121:U
1117:1
1114:U
1093:U
1087:V
1067:V
1061:U
1036:e
1032:]
1028:V
1025:[
1022:=
1017:e
1013:]
1009:U
1006:[
978:U
972:V
952:V
946:U
921:m
917:]
913:V
910:[
907:=
902:m
898:]
894:U
891:[
877:V
873:U
851:(
840:V
836:V
828:n
824:V
822:(
819:n
810:V
802:V
775:B
768:P
754:)
751:P
748:(
743:B
738:d
735:n
732:E
724:R
714:B
710:B
706:R
687:-
685:p
675:G
668:p
644:]
641:G
638:[
634:F
620:D
615:n
612:T
546:R
539:R
517:R
513:R
511:i
509:e
500:i
498:e
484:R
479:i
475:e
469:n
464:1
461:=
458:i
450:=
447:R
437:R
425:M
420:R
415:M
407:M
402:R
389:M
384:R
368:M
364:M
356:M
348:M
340:M
328:M
324:M
320:R
247:n
233:1
227:n
203:Z
199:n
195:/
190:Z
165:R
141:1
137:N
128:2
124:N
101:2
97:N
88:1
84:N
73:M
69:2
66:N
62:1
59:N
43:R
32:M
20:.
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