1542:, 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.
1482:
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
710:
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
861:
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
566:
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.
856:
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
791:
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
1483:
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
441:
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
574:. 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
1640:
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
1257:
1211:
775:
225:
505:
164:
124:
1630:
186:
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.
1114:
1088:
999:
973:
804:
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
665:
1469:
1394:
1059:
944:
1295:
254:
1315:
559:
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
305:
Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side (for example, uniserial, serial,
1553:
to refer to an
Artinian serial ring. Nakayama showed that all modules over such rings are serial. Artinian serial rings are sometimes called
1538:
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
1823:
278:
A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen,
1534:(literally "one-series") during investigations of rings over which all modules are direct sums of cyclic submodules. For this reason,
1907:
1889:
1868:
1841:
1805:
1546:
313:) then it is assumed the condition holds on both the left and right. Unless otherwise specified, each ring in this article is a
299:
2025:
Příhoda, Pavel (2004), "Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension",
562:
The most general result, perhaps, on the modules of a serial ring is attributed to Drozd and
Warfield: it states that every
2089:
Puninski, Gennadi (2001b), "Some model theory over a nearly simple uniserial domain and decompositions of serial modules",
866:). To paraphrase the result as it appears there: A basic Artinian serial ring whose quiver is a circle is a homomorphic
2111:
Puninski, Gennadi (2001c), "Some model theory over an exceptional uniserial ring and decompositions of serial modules",
2047:
Příhoda, Pavel (2006), "A version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules",
1216:
1170:
730:
563:
195:
675:
515:
453:
582:
616:
2162:
1967:
Köthe, Gottfried (1935), "Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring. (German)",
129:
89:
65:
1575:
1120:
1002:
373:
and thus is directly indecomposable. It is also easy to see that every finitely generated submodule of
346:
171:
2167:
1508:
871:
842:
809:
801:
538:
46:
1093:
1067:
978:
952:
867:
682:
608:
is a uniserial ring, and all Artinian principal ideal rings are serial rings, as is illustrated by
342:
639:
362:
2014:
1877:
1539:
1260:
50:
1564:
for a serial module with the additional property that for any two finitely generated submodules
1527:
570:
Being right serial is preserved under direct products of rings and modules, and preserved under
295:
1399:
1324:
1012:
897:
1903:
1885:
1864:
1837:
1819:
1801:
712:
598:
389:
354:
1280:
2140:
2120:
2098:
2076:
2056:
2034:
2004:
1996:
1976:
1954:
1932:
1637:
1554:
1484:
858:
816:
546:
542:
531:
291:
233:
35:
1300:
793:
692:
609:
314:
1504:
699:
605:
604:
Many examples of serial rings can be gleaned from the structure sections above. Every
553:
403:
370:
283:
2103:
2156:
2145:
1520:
620:
594:
571:
421:
318:
306:
287:
190:
2039:
947:
782:
689:
672:
668:
1959:
1795:
2131:
Warfield, Robert B. Jr. (1975), "Serial rings and finitely presented modules.",
1275:
1062:
805:
788:. This is why the results are phrased in terms of indecomposable, basic rings.
261:
61:
2124:
2081:
2060:
2009:
1523:
has "chain" as its namesake, but it is in general not related to chain rings.
797:
634:
407:
310:
17:
1937:
1923:
Eisenbud, David; Griffith, Phillip (1971), "The structure of serial rings",
279:
57:
28:
1834:
Endomorphism rings and direct sum decompositions in some classes of modules
1644:
is a finite direct sum of homogeneously serial right ideals if and only if
1863:, Mathematical Surveys and Monographs, 65. American Mathematical Society,
1861:
Rings and things and a fine array of twentieth century associative algebra
1854:, Grundlehren der Mathematischen Wissenschaften, No. 191. Springer-Verlag
182:
if it is uniserial as a right module over itself, and likewise called a
2018:
1980:
824:
271:
has been used differently from the above definition: for clarification
228:
2000:
1945:
Facchini, Alberto (1996), "Krull-Schmidt fails for serial modules",
1656:
matrix ring over a local serial ring. Such rings are also known as
272:
2067:
Puninski, G. T. (2002), "Artinian and Noetherian serial rings.",
541:(which are a special case of serial rings) are direct sums of
1005:
notion can be defined: the modules are said to have the same
534:, which is a stronger condition than being a semilocal ring.
1777:
1681:
1987:
Nakayama, Tadasi (1941), "On Frobeniusean algebras. II.",
711:
semiperfect rings such as serial rings, the basic ring is
1511:. By the same token, uniserial modules have been called
1818:, Research Notes in Mathematics, vol. 44, Pitman,
27:"Chain ring" redirects here. For the bicycle part, see
727:
is known, the theory of Morita equivalence gives that
1578:
1402:
1327:
1303:
1283:
1219:
1173:
1096:
1070:
1015:
981:
955:
900:
733:
642:
578:
direct sums of uniserial modules are serial modules.
456:
256:. This ring is always serial, and is uniserial when
236:
198:
132:
92:
325:
Properties of uniserial and serial rings and modules
526:is a local, uniserial module. This indicates that
1705:
1624:
1463:
1388:
1309:
1289:
1251:
1205:
1108:
1082:
1053:
993:
967:
938:
870:of a "blow-up" of a basic, indecomposable, serial
769:
659:
499:
248:
219:
158:
118:
1507:, which are by definition commutative, uniserial
1495:Right uniserial rings can also be referred to as
428:is assumed to be Artinian or Noetherian, then End
68:. This means simply that for any two submodules
1557:, and they have a well-developed module theory.
545:. Later, Cohen and Kaplansky determined that a
1880:; Gubareni, Nadiya; Kirichenko, V. V. (2004),
1252:{\displaystyle V_{1}\oplus \dots \oplus V_{t}}
1206:{\displaystyle U_{1}\oplus \dots \oplus U_{n}}
849:on and above the diagonal, and entries from J(
1491:Notes on alternate, similar and related terms
1163:non-zero uniserial right modules over a ring
377:can be generated by a single element, and so
8:
1682:Hazewinkel, Gubareni & Kirichenko (2004)
770:{\displaystyle R\cong \mathrm {End} _{B}(P)}
302:, P. Příhoda, G. Puninski, and R. Warfield.
1676:References for each author can be found in
2113:Journal of the London Mathematical Society
1814:Chatters, A. W.; Hajarnavis, C.R. (1980),
1794:Frank W. Anderson; Kent R. Fuller (1992),
1778:Hazewinkel, Gubareni & Kirichenko 2004
537:Köthe showed that the modules of Artinian
220:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
2144:
2102:
2080:
2038:
2008:
1958:
1936:
1677:
1605:
1582:
1577:
1455:
1436:
1420:
1410:
1401:
1380:
1361:
1345:
1335:
1326:
1302:
1282:
1243:
1224:
1218:
1197:
1178:
1172:
1095:
1069:
1045:
1026:
1014:
980:
954:
930:
911:
899:
752:
741:
732:
644:
643:
641:
500:{\displaystyle R=\oplus _{i=1}^{n}e_{i}R}
488:
478:
467:
455:
235:
213:
212:
204:
200:
199:
197:
150:
137:
131:
110:
97:
91:
1753:
1741:
863:
1693:
1669:
1765:
1729:
1717:
823:) is nonzero. This matrix ring is a
597:is trivially uniserial, and likewise
7:
1882:Algebras, rings and modules. Vol. 1.
1530:and Keizo Asano introduced the term
329:It is immediate that in a uniserial
159:{\displaystyle N_{2}\subseteq N_{1}}
119:{\displaystyle N_{1}\subseteq N_{2}}
878:A decomposition uniqueness property
1625:{\displaystyle A/J(A)\cong B/J(B)}
1487:, couniformly presented modules.)
748:
745:
742:
585:holds in Noetherian serial rings.
552:has this property for its modules
189:An easy motivating example is the
25:
1658:primary decomposable serial rings
719:is a serial ring with basic ring
615:More exotic examples include the
450:necessarily factors in the form
1797:Rings and Categories of Modules
1503:. This latter term alludes to
1119:The following weak form of the
715:to the original ring. Thus if
2040:10.1016/j.jalgebra.2004.06.027
1902:, Kluwer Academic Publishers,
1884:, Kluwer Academic Publishers,
1800:, Springer, pp. 347–349,
1706:Chatters & Hajarnavis 1980
1619:
1613:
1596:
1590:
1452:
1446:
1440:
1429:
1417:
1403:
1377:
1371:
1365:
1354:
1342:
1328:
1109:{\displaystyle V\rightarrow U}
1100:
1083:{\displaystyle U\rightarrow V}
1074:
1042:
1035:
1023:
1016:
994:{\displaystyle V\rightarrow U}
985:
968:{\displaystyle U\rightarrow V}
959:
927:
920:
908:
901:
764:
758:
654:
648:
174:of uniserial modules. A ring
1:
2104:10.1016/s0022-4049(00)00140-7
1960:10.1090/s0002-9947-96-01740-0
808:over a Noetherian, uniserial
2146:10.1016/0021-8693(75)90074-5
660:{\displaystyle \mathbb {F} }
1898:Puninski, Gennadi (2001a),
1816:Rings with chain conditions
1562:homogeneously serial module
1545:Expanding on Köthe's work,
781:is some finitely generated
446:rings. A right serial ring
2184:
1832:Facchini, Alberto (1998),
1551:generalized uniserial ring
890:are said to have the same
581:It has been verified that
298:, H. Kuppisch, I. Murase,
26:
2125:10.1112/s0024610701002344
2061:10.1080/00927870500455049
1852:Algebra. II. Ring theory.
1464:{\displaystyle _{e}=_{e}}
1389:{\displaystyle _{m}=_{m}}
1054:{\displaystyle _{e}=_{e}}
939:{\displaystyle _{m}=_{m}}
617:upper triangular matrices
406:which is very close to a
341:and 0 are simultaneously
2069:J. Math. Sci. (New York)
1648:is isomorphic to a full
1266:-modules if and only if
337:, all submodules except
166:. A module is called a
2082:10.1023/A:1014906008243
1947:Trans. Amer. Math. Soc.
1938:10.2140/pjm.1971.36.109
1560:Warfield used the term
1290:{\displaystyle \sigma }
1167:. Then the direct sums
723:, and the structure of
1626:
1465:
1390:
1311:
1291:
1253:
1207:
1110:
1084:
1055:
995:
969:
940:
771:
661:
501:
250:
249:{\displaystyle n>1}
221:
160:
120:
2091:J. Pure Appl. Algebra
1989:Annals of Mathematics
1836:, Birkhäuser Verlag,
1627:
1515:, and serial modules
1501:right valuation rings
1466:
1391:
1312:
1310:{\displaystyle \tau }
1292:
1254:
1208:
1121:Krull-Schmidt theorem
1111:
1085:
1061:, if there exists an
1056:
996:
970:
941:
772:
662:
583:Jacobson's conjecture
539:principal ideal rings
502:
410:in the sense that End
388:It is known that the
317:, and each module is
251:
222:
161:
121:
1859:Faith, Carl (1999),
1850:Faith, Carl (1976),
1576:
1400:
1325:
1301:
1281:
1274:and there exist two
1217:
1171:
1094:
1068:
1013:
979:
953:
946:, if there exists a
898:
872:quasi-Frobenius ring
731:
640:
601:are serial modules.
454:
422:maximal right ideals
290:, Phillip Griffith,
234:
196:
180:right uniserial ring
130:
90:
1878:Hazewinkel, Michiel
1519:. The notion of a
1090:and an epimorphism
975:and a monomorphism
483:
2010:10338.dmlcz/140501
1981:10.1007/bf01201343
1622:
1540:composition series
1461:
1386:
1307:
1287:
1249:
1203:
1106:
1080:
1051:
991:
965:
936:
845:with entries from
841:, and consists of
767:
657:
599:semisimple modules
572:quotients of rings
564:finitely presented
516:idempotent element
497:
463:
369:is also clearly a
294:, V.V Kirichenko,
246:
217:
156:
116:
1991:, Second Series,
1953:(11): 4561–4575,
1825:978-0-273-08446-4
1555:Nakayama algebras
1517:semichain modules
1497:right chain rings
1485:injective modules
713:Morita equivalent
543:cyclic submodules
438:is a local ring.
390:endomorphism ring
355:maximal submodule
184:right serial ring
16:(Redirected from
2175:
2149:
2148:
2127:
2107:
2106:
2085:
2084:
2063:
2055:(4): 1479–1487,
2043:
2042:
2021:
2012:
1983:
1963:
1962:
1941:
1940:
1925:Pacific J. Math.
1912:
1894:
1873:
1855:
1846:
1828:
1810:
1781:
1775:
1769:
1763:
1757:
1751:
1745:
1739:
1733:
1727:
1721:
1715:
1709:
1703:
1697:
1691:
1685:
1678:Puninski (2001a)
1674:
1638:Jacobson radical
1636:(−) denotes the
1631:
1629:
1628:
1623:
1609:
1586:
1547:Tadashi Nakayama
1470:
1468:
1467:
1462:
1460:
1459:
1450:
1449:
1425:
1424:
1415:
1414:
1395:
1393:
1392:
1387:
1385:
1384:
1375:
1374:
1350:
1349:
1340:
1339:
1316:
1314:
1313:
1308:
1296:
1294:
1293:
1288:
1258:
1256:
1255:
1250:
1248:
1247:
1229:
1228:
1212:
1210:
1209:
1204:
1202:
1201:
1183:
1182:
1115:
1113:
1112:
1107:
1089:
1087:
1086:
1081:
1060:
1058:
1057:
1052:
1050:
1049:
1031:
1030:
1000:
998:
997:
992:
974:
972:
971:
966:
945:
943:
942:
937:
935:
934:
916:
915:
817:Jacobson radical
776:
774:
773:
768:
757:
756:
751:
666:
664:
663:
658:
647:
610:semisimple rings
547:commutative ring
532:semiperfect ring
506:
504:
503:
498:
493:
492:
482:
477:
420:has at most two
255:
253:
252:
247:
226:
224:
223:
218:
216:
208:
203:
165:
163:
162:
157:
155:
154:
142:
141:
125:
123:
122:
117:
115:
114:
102:
101:
40:uniserial module
36:abstract algebra
21:
2183:
2182:
2178:
2177:
2176:
2174:
2173:
2172:
2153:
2152:
2130:
2110:
2088:
2066:
2046:
2024:
2001:10.2307/1968984
1986:
1966:
1944:
1922:
1919:
1917:Primary Sources
1910:
1897:
1892:
1876:
1871:
1858:
1849:
1844:
1831:
1826:
1813:
1808:
1793:
1790:
1785:
1784:
1776:
1772:
1764:
1760:
1752:
1748:
1740:
1736:
1728:
1724:
1716:
1712:
1704:
1700:
1692:
1688:
1675:
1671:
1666:
1652: ×
1574:
1573:
1528:Gottfried Köthe
1505:valuation rings
1493:
1451:
1432:
1416:
1406:
1398:
1397:
1376:
1357:
1341:
1331:
1323:
1322:
1299:
1298:
1279:
1278:
1239:
1220:
1215:
1214:
1193:
1174:
1169:
1168:
1154:
1145:
1138:
1129:
1092:
1091:
1066:
1065:
1041:
1022:
1011:
1010:
977:
976:
951:
950:
926:
907:
896:
895:
880:
832:
740:
729:
728:
708:
638:
637:
628:
591:
523:
512:
484:
452:
451:
433:
415:
397:
327:
315:ring with unity
286:, A. Facchini,
232:
231:
194:
193:
146:
133:
128:
127:
106:
93:
88:
87:
81:
74:
62:totally ordered
32:
23:
22:
15:
12:
11:
5:
2181:
2179:
2171:
2170:
2165:
2155:
2154:
2151:
2150:
2139:(2): 187–222,
2128:
2119:(2): 311–326,
2108:
2097:(3): 319–337,
2086:
2064:
2044:
2022:
1984:
1964:
1942:
1918:
1915:
1914:
1913:
1908:
1895:
1890:
1874:
1869:
1856:
1847:
1842:
1829:
1824:
1811:
1806:
1789:
1786:
1783:
1782:
1770:
1758:
1746:
1734:
1722:
1710:
1698:
1686:
1668:
1667:
1665:
1662:
1621:
1618:
1615:
1612:
1608:
1604:
1601:
1598:
1595:
1592:
1589:
1585:
1581:
1549:used the term
1526:In the 1930s,
1492:
1489:
1458:
1454:
1448:
1445:
1442:
1439:
1435:
1431:
1428:
1423:
1419:
1413:
1409:
1405:
1383:
1379:
1373:
1370:
1367:
1364:
1360:
1356:
1353:
1348:
1344:
1338:
1334:
1330:
1317:of 1, 2, ...,
1306:
1286:
1246:
1242:
1238:
1235:
1232:
1227:
1223:
1200:
1196:
1192:
1189:
1186:
1181:
1177:
1150:
1143:
1134:
1127:
1105:
1102:
1099:
1079:
1076:
1073:
1048:
1044:
1040:
1037:
1034:
1029:
1025:
1021:
1018:
990:
987:
984:
964:
961:
958:
933:
929:
925:
922:
919:
914:
910:
906:
903:
892:monogeny class
879:
876:
828:
796:, Noetherian,
766:
763:
760:
755:
750:
747:
744:
739:
736:
707:
704:
700:Sylow subgroup
676:characteristic
656:
653:
650:
646:
624:
606:valuation ring
590:
587:
554:if and only if
521:
510:
496:
491:
487:
481:
476:
473:
470:
466:
462:
459:
429:
411:
404:semilocal ring
393:
371:uniform module
326:
323:
245:
242:
239:
215:
211:
207:
202:
153:
149:
145:
140:
136:
113:
109:
105:
100:
96:
79:
72:
24:
18:Uniserial ring
14:
13:
10:
9:
6:
4:
3:
2:
2180:
2169:
2166:
2164:
2163:Module theory
2161:
2160:
2158:
2147:
2142:
2138:
2134:
2129:
2126:
2122:
2118:
2114:
2109:
2105:
2100:
2096:
2092:
2087:
2083:
2078:
2075:: 2330–2347,
2074:
2070:
2065:
2062:
2058:
2054:
2050:
2049:Comm. Algebra
2045:
2041:
2036:
2032:
2028:
2023:
2020:
2016:
2011:
2006:
2002:
1998:
1994:
1990:
1985:
1982:
1978:
1974:
1970:
1965:
1961:
1956:
1952:
1948:
1943:
1939:
1934:
1930:
1926:
1921:
1920:
1916:
1911:
1909:0-7923-7187-9
1905:
1901:
1896:
1893:
1891:1-4020-2690-0
1887:
1883:
1879:
1875:
1872:
1870:0-8218-0993-8
1866:
1862:
1857:
1853:
1848:
1845:
1843:3-7643-5908-0
1839:
1835:
1830:
1827:
1821:
1817:
1812:
1809:
1807:0-387-97845-3
1803:
1799:
1798:
1792:
1791:
1787:
1779:
1774:
1771:
1767:
1762:
1759:
1755:
1754:Warfield 1975
1750:
1747:
1743:
1742:Nakayama 1941
1738:
1735:
1731:
1726:
1723:
1719:
1714:
1711:
1707:
1702:
1699:
1695:
1690:
1687:
1683:
1679:
1673:
1670:
1663:
1661:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1616:
1610:
1606:
1602:
1599:
1593:
1587:
1583:
1579:
1571:
1567:
1563:
1558:
1556:
1552:
1548:
1543:
1541:
1537:
1533:
1529:
1524:
1522:
1521:catenary ring
1518:
1514:
1513:chain modules
1510:
1506:
1502:
1498:
1490:
1488:
1486:
1480:
1478:
1475:= 1, 2, ...,
1474:
1456:
1443:
1437:
1433:
1426:
1421:
1411:
1407:
1381:
1368:
1362:
1358:
1351:
1346:
1336:
1332:
1320:
1304:
1284:
1277:
1273:
1269:
1265:
1262:
1244:
1240:
1236:
1233:
1230:
1225:
1221:
1198:
1194:
1190:
1187:
1184:
1179:
1175:
1166:
1162:
1158:
1153:
1149:
1142:
1137:
1133:
1126:
1122:
1117:
1103:
1097:
1077:
1071:
1064:
1046:
1038:
1032:
1027:
1019:
1008:
1007:epigeny class
1004:
988:
982:
962:
956:
949:
931:
923:
917:
912:
904:
893:
889:
885:
877:
875:
873:
869:
865:
864:Puninski 2002
860:
854:
852:
848:
844:
840:
836:
831:
826:
822:
818:
814:
811:
807:
803:
800:, as well as
799:
795:
789:
787:
784:
780:
761:
753:
737:
734:
726:
722:
718:
714:
705:
703:
701:
697:
694:
691:
687:
684:
680:
677:
674:
670:
651:
636:
632:
627:
622:
621:division ring
618:
613:
611:
607:
602:
600:
596:
595:simple module
588:
586:
584:
579:
577:
573:
568:
565:
560:
558:
555:
551:
548:
544:
540:
535:
533:
529:
525:
517:
513:
494:
489:
485:
479:
474:
471:
468:
464:
460:
457:
449:
445:
439:
437:
432:
427:
423:
419:
414:
409:
405:
401:
396:
391:
386:
384:
383:Bézout module
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
324:
322:
320:
316:
312:
308:
303:
301:
297:
293:
289:
285:
282:, Yu. Drozd,
281:
276:
274:
270:
265:
263:
259:
243:
240:
237:
230:
209:
205:
192:
191:quotient ring
187:
185:
181:
177:
173:
169:
168:serial module
151:
147:
143:
138:
134:
111:
107:
103:
98:
94:
85:
78:
71:
67:
63:
59:
55:
52:
48:
44:
41:
37:
30:
19:
2136:
2132:
2116:
2112:
2094:
2090:
2072:
2068:
2052:
2048:
2030:
2026:
1992:
1988:
1972:
1968:
1950:
1946:
1928:
1924:
1900:Serial rings
1899:
1881:
1860:
1851:
1833:
1815:
1796:
1773:
1761:
1749:
1737:
1725:
1713:
1701:
1694:Příhoda 2004
1689:
1672:
1657:
1653:
1649:
1645:
1641:
1633:
1569:
1565:
1561:
1559:
1550:
1544:
1535:
1531:
1525:
1516:
1512:
1500:
1496:
1494:
1481:
1476:
1472:
1318:
1276:permutations
1271:
1267:
1263:
1164:
1160:
1156:
1151:
1147:
1140:
1135:
1131:
1124:
1118:
1006:
948:monomorphism
891:
887:
883:
882:Two modules
881:
855:
850:
846:
838:
834:
829:
820:
812:
790:
785:
783:progenerator
778:
724:
720:
716:
709:
695:
685:
678:
669:finite field
630:
625:
614:
603:
592:
580:
575:
569:
561:
556:
549:
536:
527:
519:
508:
447:
444:right Bézout
443:
440:
435:
430:
425:
417:
412:
399:
394:
387:
382:
378:
374:
366:
363:local module
358:
350:
338:
334:
330:
328:
304:
292:I. Kaplansky
277:
268:
266:
257:
188:
183:
179:
178:is called a
175:
167:
83:
76:
69:
53:
42:
39:
33:
2168:Ring theory
2033:: 332–341,
1995:(1): 1–21,
1931:: 109–121,
1123:holds. Let
1063:epimorphism
837:) for some
806:matrix ring
798:prime rings
507:where each
347:superfluous
300:T. Nakayama
288:A.W. Goldie
284:D. Eisenbud
262:prime power
170:if it is a
2157:Categories
2133:J. Algebra
2027:J. Algebra
1766:Faith 1976
1730:Köthe 1935
1718:Faith 1999
1471:for every
1321:such that
1261:isomorphic
1009:, denoted
894:, denoted
859:isomorphic
794:hereditary
635:group ring
633:, and the
530:is also a
408:local ring
311:Noetherian
172:direct sum
58:submodules
1975:: 31–44,
1788:Textbooks
1600:≅
1536:uniserial
1532:Einreihig
1438:τ
1363:σ
1305:τ
1285:σ
1237:⊕
1234:⋯
1231:⊕
1191:⊕
1188:⋯
1185:⊕
1101:→
1075:→
986:→
960:→
853:) below.
738:≅
706:Structure
688:having a
667:for some
465:⊕
343:essential
280:P.M. Cohn
273:see below
269:uniserial
267:The term
144:⊆
104:⊆
86:, either
66:inclusion
29:Chainring
1969:Math. Z.
843:matrices
815:, whose
589:Examples
333:-module
307:Artinian
296:G. Köthe
227:for any
56:, whose
2019:1968984
1509:domains
1146:, ...,
1130:, ...,
825:subring
802:quivers
629:
619:over a
434:
416:
398:
357:, then
229:integer
49:over a
2017:
1906:
1888:
1867:
1840:
1822:
1804:
1632:where
1001:. The
810:domain
777:where
693:normal
690:cyclic
576:finite
514:is an
424:. If
353:has a
349:. If
319:unital
47:module
2015:JSTOR
1664:Notes
868:image
683:group
673:prime
402:is a
381:is a
361:is a
260:is a
45:is a
1904:ISBN
1886:ISBN
1865:ISBN
1838:ISBN
1820:ISBN
1802:ISBN
1680:and
1568:and
1396:and
1297:and
1259:are
1213:and
1003:dual
886:and
827:of M
681:and
593:Any
518:and
345:and
241:>
75:and
60:are
51:ring
38:, a
2141:doi
2121:doi
2099:doi
2095:163
2077:doi
2073:110
2057:doi
2035:doi
2031:281
2005:hdl
1997:doi
1977:doi
1955:doi
1951:348
1933:doi
1499:or
1155:be
671:of
392:End
365:.
126:or
82:of
64:by
34:In
2159::
2137:37
2135:,
2117:64
2115:,
2093:,
2071:,
2053:34
2051:,
2029:,
2013:,
2003:,
1993:42
1973:39
1971:,
1949:,
1929:36
1927:,
1660:.
1572:,
1479:.
1270:=
1159:+
1139:,
1116:.
874:.
819:J(
702:.
612:.
385:.
321:.
309:,
275:.
264:.
2143::
2123::
2101::
2079::
2059::
2037::
2007::
1999::
1979::
1957::
1935::
1780:.
1768:.
1756:.
1744:.
1732:.
1720:.
1708:.
1696:.
1684:.
1654:n
1650:n
1646:R
1642:R
1634:J
1620:)
1617:B
1614:(
1611:J
1607:/
1603:B
1597:)
1594:A
1591:(
1588:J
1584:/
1580:A
1570:B
1566:A
1477:n
1473:i
1457:e
1453:]
1447:)
1444:i
1441:(
1434:V
1430:[
1427:=
1422:e
1418:]
1412:i
1408:U
1404:[
1382:m
1378:]
1372:)
1369:i
1366:(
1359:V
1355:[
1352:=
1347:m
1343:]
1337:i
1333:U
1329:[
1319:n
1272:t
1268:n
1264:R
1245:t
1241:V
1226:1
1222:V
1199:n
1195:U
1180:1
1176:U
1165:R
1161:t
1157:n
1152:t
1148:V
1144:1
1141:V
1136:n
1132:U
1128:1
1125:U
1104:U
1098:V
1078:V
1072:U
1047:e
1043:]
1039:V
1036:[
1033:=
1028:e
1024:]
1020:U
1017:[
989:U
983:V
963:V
957:U
932:m
928:]
924:V
921:[
918:=
913:m
909:]
905:U
902:[
888:V
884:U
862:(
851:V
847:V
839:n
835:V
833:(
830:n
821:V
813:V
786:B
779:P
765:)
762:P
759:(
754:B
749:d
746:n
743:E
735:R
725:B
721:B
717:R
698:-
696:p
686:G
679:p
655:]
652:G
649:[
645:F
631:D
626:n
623:T
557:R
550:R
528:R
524:R
522:i
520:e
511:i
509:e
495:R
490:i
486:e
480:n
475:1
472:=
469:i
461:=
458:R
448:R
436:M
431:R
426:M
418:M
413:R
400:M
395:R
379:M
375:M
367:M
359:M
351:M
339:M
335:M
331:R
258:n
244:1
238:n
214:Z
210:n
206:/
201:Z
176:R
152:1
148:N
139:2
135:N
112:2
108:N
99:1
95:N
84:M
80:2
77:N
73:1
70:N
54:R
43:M
31:.
20:)
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