703:
586:
267:
1844:
599:
1075:
494:
2158:
2125:
1480:
2160:. If a nilpotent infinitesimal is a variable tending to zero, it can be shown that any sum of terms for which it is the subject is an indefinitely small proportion of the first order term.
1233:
351:
1751:
1685:
1367:
1924:
As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition. More generally, in view of the above definitions, an operator
489:
1970:
1651:
1524:
1424:
1257:
1190:
1135:
824:
2003:
1903:
1400:
965:
304:
151:
879:
761:
732:
484:
455:
2045:
1599:
928:
2070:. More generally, the technique of microadditivity (which can used to derive theorems in physics) makes use of nilpotent or nilsquare infinitesimals and is part
186:
1942:
1870:
1619:
1571:
1551:
1500:
1444:
1304:
1280:
1163:
1108:
902:
844:
426:
406:
386:
110:
83:
60:
1725:
2375:
970:
2347:
2293:
2256:
2231:
1770:
1754:
1779:
2169:
2071:
2133:
2100:
1449:
698:{\displaystyle A={\begin{pmatrix}0&1\\0&1\end{pmatrix}},\;\;B={\begin{pmatrix}0&1\\0&0\end{pmatrix}}.}
2067:
1910:
851:
1195:
321:
157:
1730:
1660:
1283:
1309:
2063:
1947:
2365:
1632:
1505:
1405:
1238:
1171:
1142:
1116:
2370:
2018:
847:
358:
28:
931:
782:
362:
63:
1079:
More generally, the sum of a unit element and a nilpotent element is a unit when they commute.
803:
2343:
2289:
2252:
2227:
1482:. As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent
354:
32:
1975:
1875:
1372:
937:
276:
123:
2309:
2189:
2087:
2086:
contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include
1906:
1530:
1138:
1088:
307:
1533:
and annihilation of simple modules is available for nilradical: nilpotent elements of ring
857:
581:{\displaystyle {\begin{aligned}c^{2}&=(ba)^{2}\\&=b(ab)a\\&=0.\\\end{aligned}}}
262:{\displaystyle A={\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}}}
1766:
737:
708:
460:
431:
161:
24:
20:
2024:
1576:
907:
1621:). This follows from the fact that nilradical is the intersection of all prime ideals.
2091:
1927:
1855:
1774:
1604:
1556:
1536:
1485:
1429:
1289:
1265:
1148:
1093:
887:
829:
411:
391:
371:
95:
68:
45:
2313:
1690:
2359:
2056:
2048:
2006:
174:
2247:
Atiyah, M. F.; MacDonald, I. G. (February 21, 1994). "Chapter 1: Rings and Ideals".
2179:
2095:
2052:
1913:
representation for
Fermionic fields are nilpotents since their squares vanish. The
794:
786:
769:
2066:
of a plane wave without sources is nilpotent when it is expressed in terms of the
1773:, which transform from one state to another, for example the raising and lowering
1402:. The prime ideals of the localized ring correspond exactly to those prime ideals
2083:
1914:
1654:
1166:
316:
39:
2010:
1553:
are precisely those that annihilate all integral domains internal to the ring
1111:
19:
This article is about a type of element in a ring. For the type of group, see
2184:
2174:
2128:
2017:
it has a nilpotent matrix in some basis. Another example for this is the
1918:
1850:
90:
2014:
1839:{\displaystyle \sigma _{\pm }=(\sigma _{x}\pm i\sigma _{y})/2}
164:
in the context of his work on the classification of algebras.
2222:
Matsumura, Hideyuki (1970). "Chapter 1: Elementary
Results".
1769:
in a finite dimensional space is nilpotent. They represent
1070:{\displaystyle (1-x)(1+x+x^{2}+\cdots +x^{n-1})=1-x^{n}=1.}
2288:. Algebras and applications, Volume 1. Springer, 2002.
1259:
is contained in the intersection of all prime ideals.
661:
614:
201:
2136:
2103:
2027:
1978:
1950:
1930:
1878:
1858:
1782:
1733:
1693:
1663:
1635:
1607:
1579:
1559:
1539:
1508:
1488:
1452:
1432:
1408:
1375:
1312:
1292:
1268:
1241:
1198:
1174:
1151:
1119:
1096:
973:
940:
910:
890:
860:
832:
806:
740:
711:
602:
492:
463:
434:
414:
394:
374:
324:
279:
189:
126:
98:
71:
48:
2152:
2119:
2039:
1997:
1964:
1936:
1897:
1864:
1838:
1745:
1719:
1679:
1645:
1613:
1593:
1565:
1545:
1518:
1494:
1474:
1438:
1418:
1394:
1361:
1298:
1274:
1251:
1227:
1184:
1157:
1129:
1102:
1069:
959:
922:
896:
873:
838:
818:
755:
726:
697:
580:
478:
449:
420:
400:
380:
345:
298:
261:
145:
104:
77:
54:
1526:is exactly the intersection of all prime ideals.
2153:{\displaystyle \mathbb {C} \otimes \mathbb {O} }
2120:{\displaystyle \mathbb {C} \otimes \mathbb {H} }
1475:{\displaystyle {\mathfrak {p}}\cap S=\emptyset }
173:This definition can be applied in particular to
8:
2340:Zero to Infinity: The Foundations of Physics
2284:Polcino Milies, CĂ©sar; Sehgal, Sudarshan K.
1356:
1319:
1165:in a commutative ring is contained in every
2308:, Class. Quantum Grav. 17:3703–3714, 2000
1502:is not contained in some prime ideal. Thus
1228:{\displaystyle x^{n}=0\in {\mathfrak {p}}}
649:
648:
346:{\displaystyle \mathbb {Z} /9\mathbb {Z} }
2306:The topological particle and Morse theory
2146:
2145:
2138:
2137:
2135:
2113:
2112:
2105:
2104:
2102:
2026:
1983:
1977:
1958:
1957:
1949:
1929:
1883:
1877:
1857:
1828:
1819:
1803:
1787:
1781:
1753:is a nilpotent transformation. See also:
1732:
1708:
1707:
1698:
1697:
1692:
1671:
1670:
1662:
1637:
1636:
1634:
1606:
1583:
1578:
1558:
1538:
1510:
1509:
1507:
1487:
1454:
1453:
1451:
1431:
1410:
1409:
1407:
1380:
1374:
1338:
1311:
1291:
1267:
1243:
1242:
1240:
1219:
1218:
1203:
1197:
1176:
1175:
1173:
1150:
1121:
1120:
1118:
1095:
1055:
1027:
1008:
972:
945:
939:
909:
889:
865:
859:
831:
805:
739:
710:
656:
609:
601:
527:
501:
493:
491:
462:
433:
413:
393:
373:
339:
338:
330:
326:
325:
323:
284:
278:
196:
188:
131:
125:
97:
70:
47:
2201:
16:Element in a ring whose some power is 0
1755:Jordan decomposition in a Lie algebra
1145:of the ring. Every nilpotent element
7:
2208:Polcino Milies & Sehgal (2002),
1746:{\displaystyle \operatorname {ad} x}
1680:{\displaystyle x\in {\mathfrak {g}}}
1529:A characteristic similar to that of
2249:Introduction to Commutative Algebra
1771:creation and annihilation operators
1709:
1699:
1672:
1638:
1511:
1455:
1411:
1362:{\displaystyle S=\{1,x,x^{2},...\}}
1244:
1220:
1177:
1122:
1469:
789:, which has only a single element
14:
2342:, London, World Scientific 2007,
2047:). Both are linked, also through
1965:{\displaystyle n\in \mathbb {N} }
1625:Nilpotent elements in Lie algebra
1282:is not nilpotent, we are able to
27:. For the type of semigroup, see
2376:Algebraic properties of elements
2170:Idempotent element (ring theory)
1687:is called nilpotent if it is in
850:is nilpotent if and only if its
768:By definition, any element of a
156:The term, along with its sister
1646:{\displaystyle {\mathfrak {g}}}
1519:{\displaystyle {\mathfrak {N}}}
1419:{\displaystyle {\mathfrak {p}}}
1252:{\displaystyle {\mathfrak {N}}}
1185:{\displaystyle {\mathfrak {p}}}
1137:; this is a consequence of the
1130:{\displaystyle {\mathfrak {N}}}
357:of 3 is nilpotent because 3 is
31:. For the type of algebra, see
2329:. J.Diff.Geom.17:661–692,1982.
2327:Supersymmetry and Morse theory
2286:An introduction to group rings
2210:An Introduction to Group Rings
1825:
1796:
1714:
1694:
1286:with respect to the powers of
1087:The nilpotent elements from a
1039:
989:
986:
974:
793:). All nilpotent elements are
781:No nilpotent element can be a
588:An example with matrices (for
555:
546:
524:
514:
89:if there exists some positive
1:
2251:. Westview Press. p. 5.
2226:. W. A. Benjamin. p. 6.
2072:smooth infinitesimal analysis
23:. For the type of ideal, see
2314:10.1088/0264-9381/17/18/309
1917:is an important example in
2392:
2273:Linear Associative Algebra
18:
2068:algebra of physical space
2059:in a celebrated article.
1944:is nilpotent if there is
852:characteristic polynomial
819:{\displaystyle n\times n}
368:Assume that two elements
1998:{\displaystyle Q^{n}=0}
1898:{\displaystyle Q^{2}=0}
1395:{\displaystyle S^{-1}R}
1369:to get a non-zero ring
960:{\displaystyle x^{n}=0}
299:{\displaystyle A^{3}=0}
146:{\displaystyle x^{n}=0}
2154:
2121:
2041:
1999:
1966:
1938:
1899:
1866:
1840:
1747:
1721:
1681:
1647:
1615:
1595:
1573:(that is, of the form
1567:
1547:
1520:
1496:
1476:
1440:
1420:
1396:
1363:
1300:
1276:
1253:
1229:
1186:
1159:
1131:
1104:
1071:
961:
924:
898:
875:
840:
820:
757:
728:
699:
582:
480:
451:
422:
402:
382:
347:
300:
263:
147:
106:
79:
56:
2155:
2122:
2064:electromagnetic field
2042:
2000:
1967:
1939:
1900:
1867:
1841:
1761:Nilpotency in physics
1748:
1722:
1682:
1648:
1616:
1596:
1568:
1548:
1521:
1497:
1477:
1441:
1421:
1397:
1364:
1301:
1277:
1254:
1230:
1187:
1160:
1132:
1105:
1072:
962:
925:
899:
876:
874:{\displaystyle t^{n}}
841:
821:
758:
729:
700:
583:
481:
452:
423:
403:
383:
348:
301:
273:is nilpotent because
264:
148:
107:
80:
57:
2134:
2101:
2082:The two-dimensional
2078:Algebraic nilpotents
2025:
1976:
1948:
1928:
1876:
1856:
1780:
1731:
1691:
1661:
1633:
1605:
1577:
1557:
1537:
1506:
1486:
1450:
1430:
1406:
1373:
1310:
1290:
1266:
1239:
1196:
1192:of that ring, since
1172:
1149:
1141:. This ideal is the
1117:
1094:
971:
938:
908:
888:
858:
846:with entries from a
830:
804:
756:{\displaystyle BA=B}
738:
727:{\displaystyle AB=0}
709:
600:
490:
479:{\displaystyle c=ba}
461:
450:{\displaystyle ab=0}
432:
412:
392:
372:
322:
277:
187:
160:, was introduced by
124:
96:
69:
46:
2224:Commutative Algebra
2040:{\displaystyle n=2}
2019:exterior derivative
1594:{\displaystyle R/I}
923:{\displaystyle 1-x}
904:is nilpotent, then
457:. Then the element
29:Nilpotent semigroup
2150:
2117:
2037:
1995:
1962:
1934:
1895:
1862:
1836:
1743:
1717:
1677:
1657:. Then an element
1643:
1611:
1591:
1563:
1543:
1516:
1492:
1472:
1436:
1416:
1392:
1359:
1296:
1272:
1249:
1225:
1182:
1155:
1127:
1100:
1067:
957:
920:
894:
871:
836:
816:
753:
724:
695:
686:
639:
578:
576:
476:
447:
418:
398:
378:
343:
296:
259:
253:
143:
116:(or sometimes the
102:
75:
52:
2348:978-981-270-914-1
2294:978-1-4020-0238-0
2258:978-0-201-40751-8
2233:978-0-805-37025-6
2090:(coquaternions),
2088:split-quaternions
1937:{\displaystyle Q}
1907:Grassmann numbers
1865:{\displaystyle Q}
1614:{\displaystyle I}
1601:for prime ideals
1566:{\displaystyle R}
1546:{\displaystyle R}
1495:{\displaystyle x}
1439:{\displaystyle R}
1299:{\displaystyle x}
1275:{\displaystyle x}
1158:{\displaystyle x}
1103:{\displaystyle R}
1083:Commutative rings
897:{\displaystyle x}
839:{\displaystyle A}
421:{\displaystyle R}
401:{\displaystyle b}
381:{\displaystyle a}
355:equivalence class
105:{\displaystyle n}
78:{\displaystyle R}
55:{\displaystyle x}
33:Nilpotent algebra
2383:
2350:
2336:
2330:
2323:
2317:
2302:
2296:
2282:
2276:
2269:
2263:
2262:
2244:
2238:
2237:
2219:
2213:
2206:
2190:Nilpotent matrix
2159:
2157:
2156:
2151:
2149:
2141:
2126:
2124:
2123:
2118:
2116:
2108:
2046:
2044:
2043:
2038:
2004:
2002:
2001:
1996:
1988:
1987:
1971:
1969:
1968:
1963:
1961:
1943:
1941:
1940:
1935:
1904:
1902:
1901:
1896:
1888:
1887:
1871:
1869:
1868:
1863:
1845:
1843:
1842:
1837:
1832:
1824:
1823:
1808:
1807:
1792:
1791:
1752:
1750:
1749:
1744:
1726:
1724:
1723:
1720:{\displaystyle }
1718:
1713:
1712:
1703:
1702:
1686:
1684:
1683:
1678:
1676:
1675:
1652:
1650:
1649:
1644:
1642:
1641:
1620:
1618:
1617:
1612:
1600:
1598:
1597:
1592:
1587:
1572:
1570:
1569:
1564:
1552:
1550:
1549:
1544:
1531:Jacobson radical
1525:
1523:
1522:
1517:
1515:
1514:
1501:
1499:
1498:
1493:
1481:
1479:
1478:
1473:
1459:
1458:
1445:
1443:
1442:
1437:
1425:
1423:
1422:
1417:
1415:
1414:
1401:
1399:
1398:
1393:
1388:
1387:
1368:
1366:
1365:
1360:
1343:
1342:
1305:
1303:
1302:
1297:
1281:
1279:
1278:
1273:
1258:
1256:
1255:
1250:
1248:
1247:
1234:
1232:
1231:
1226:
1224:
1223:
1208:
1207:
1191:
1189:
1188:
1183:
1181:
1180:
1164:
1162:
1161:
1156:
1139:binomial theorem
1136:
1134:
1133:
1128:
1126:
1125:
1109:
1107:
1106:
1101:
1089:commutative ring
1076:
1074:
1073:
1068:
1060:
1059:
1038:
1037:
1013:
1012:
966:
964:
963:
958:
950:
949:
929:
927:
926:
921:
903:
901:
900:
895:
880:
878:
877:
872:
870:
869:
845:
843:
842:
837:
825:
823:
822:
817:
792:
762:
760:
759:
754:
733:
731:
730:
725:
704:
702:
701:
696:
691:
690:
644:
643:
587:
585:
584:
579:
577:
564:
536:
532:
531:
506:
505:
486:is nilpotent as
485:
483:
482:
477:
456:
454:
453:
448:
427:
425:
424:
419:
407:
405:
404:
399:
387:
385:
384:
379:
352:
350:
349:
344:
342:
334:
329:
308:nilpotent matrix
305:
303:
302:
297:
289:
288:
268:
266:
265:
260:
258:
257:
152:
150:
149:
144:
136:
135:
111:
109:
108:
103:
84:
82:
81:
76:
61:
59:
58:
53:
2391:
2390:
2386:
2385:
2384:
2382:
2381:
2380:
2356:
2355:
2354:
2353:
2337:
2333:
2324:
2320:
2303:
2299:
2283:
2279:
2270:
2266:
2259:
2246:
2245:
2241:
2234:
2221:
2220:
2216:
2207:
2203:
2198:
2166:
2132:
2131:
2099:
2098:
2092:split-octonions
2080:
2023:
2022:
1979:
1974:
1973:
1946:
1945:
1926:
1925:
1879:
1874:
1873:
1872:that satisfies
1854:
1853:
1815:
1799:
1783:
1778:
1777:
1767:ladder operator
1763:
1729:
1728:
1689:
1688:
1659:
1658:
1631:
1630:
1627:
1603:
1602:
1575:
1574:
1555:
1554:
1535:
1534:
1504:
1503:
1484:
1483:
1448:
1447:
1428:
1427:
1404:
1403:
1376:
1371:
1370:
1334:
1308:
1307:
1288:
1287:
1264:
1263:
1237:
1236:
1199:
1194:
1193:
1170:
1169:
1147:
1146:
1115:
1114:
1092:
1091:
1085:
1051:
1023:
1004:
969:
968:
941:
936:
935:
906:
905:
886:
885:
861:
856:
855:
828:
827:
802:
801:
790:
785:(except in the
779:
736:
735:
707:
706:
685:
684:
679:
673:
672:
667:
657:
638:
637:
632:
626:
625:
620:
610:
598:
597:
575:
574:
562:
561:
534:
533:
523:
507:
497:
488:
487:
459:
458:
430:
429:
410:
409:
390:
389:
370:
369:
320:
319:
280:
275:
274:
252:
251:
246:
241:
235:
234:
229:
224:
218:
217:
212:
207:
197:
185:
184:
175:square matrices
170:
162:Benjamin Peirce
127:
122:
121:
94:
93:
67:
66:
44:
43:
36:
25:Nilpotent ideal
21:Nilpotent group
17:
12:
11:
5:
2389:
2387:
2379:
2378:
2373:
2368:
2358:
2357:
2352:
2351:
2331:
2318:
2297:
2277:
2264:
2257:
2239:
2232:
2214:
2200:
2199:
2197:
2194:
2193:
2192:
2187:
2182:
2177:
2172:
2165:
2162:
2148:
2144:
2140:
2127:, and complex
2115:
2111:
2107:
2079:
2076:
2055:, as shown by
2036:
2033:
2030:
1994:
1991:
1986:
1982:
1960:
1956:
1953:
1933:
1909:which allow a
1905:is nilpotent.
1894:
1891:
1886:
1882:
1861:
1835:
1831:
1827:
1822:
1818:
1814:
1811:
1806:
1802:
1798:
1795:
1790:
1786:
1775:Pauli matrices
1762:
1759:
1742:
1739:
1736:
1716:
1711:
1706:
1701:
1696:
1674:
1669:
1666:
1640:
1626:
1623:
1610:
1590:
1586:
1582:
1562:
1542:
1513:
1491:
1471:
1468:
1465:
1462:
1457:
1435:
1413:
1391:
1386:
1383:
1379:
1358:
1355:
1352:
1349:
1346:
1341:
1337:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1295:
1271:
1246:
1222:
1217:
1214:
1211:
1206:
1202:
1179:
1154:
1124:
1099:
1084:
1081:
1066:
1063:
1058:
1054:
1050:
1047:
1044:
1041:
1036:
1033:
1030:
1026:
1022:
1019:
1016:
1011:
1007:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
956:
953:
948:
944:
919:
916:
913:
893:
868:
864:
835:
815:
812:
809:
778:
775:
774:
773:
765:
764:
752:
749:
746:
743:
723:
720:
717:
714:
694:
689:
683:
680:
678:
675:
674:
671:
668:
666:
663:
662:
660:
655:
652:
647:
642:
636:
633:
631:
628:
627:
624:
621:
619:
616:
615:
613:
608:
605:
573:
570:
567:
565:
563:
560:
557:
554:
551:
548:
545:
542:
539:
537:
535:
530:
526:
522:
519:
516:
513:
510:
508:
504:
500:
496:
495:
475:
472:
469:
466:
446:
443:
440:
437:
417:
397:
377:
366:
341:
337:
333:
328:
312:
311:
295:
292:
287:
283:
271:
270:
269:
256:
250:
247:
245:
242:
240:
237:
236:
233:
230:
228:
225:
223:
220:
219:
216:
213:
211:
208:
206:
203:
202:
200:
195:
192:
179:
178:
169:
166:
142:
139:
134:
130:
101:
74:
51:
15:
13:
10:
9:
6:
4:
3:
2:
2388:
2377:
2374:
2372:
2369:
2367:
2364:
2363:
2361:
2349:
2345:
2341:
2338:Rowlands, P.
2335:
2332:
2328:
2322:
2319:
2315:
2311:
2307:
2301:
2298:
2295:
2291:
2287:
2281:
2278:
2274:
2268:
2265:
2260:
2254:
2250:
2243:
2240:
2235:
2229:
2225:
2218:
2215:
2211:
2205:
2202:
2195:
2191:
2188:
2186:
2183:
2181:
2178:
2176:
2173:
2171:
2168:
2167:
2163:
2161:
2142:
2130:
2109:
2097:
2096:biquaternions
2093:
2089:
2085:
2077:
2075:
2073:
2069:
2065:
2060:
2058:
2057:Edward Witten
2054:
2050:
2049:supersymmetry
2034:
2031:
2028:
2020:
2016:
2013:is nilpotent
2012:
2008:
2007:zero function
1992:
1989:
1984:
1980:
1954:
1951:
1931:
1922:
1920:
1916:
1912:
1911:path integral
1908:
1892:
1889:
1884:
1880:
1859:
1852:
1847:
1833:
1829:
1820:
1816:
1812:
1809:
1804:
1800:
1793:
1788:
1784:
1776:
1772:
1768:
1760:
1758:
1756:
1740:
1737:
1734:
1704:
1667:
1664:
1656:
1624:
1622:
1608:
1588:
1584:
1580:
1560:
1540:
1532:
1527:
1489:
1466:
1463:
1460:
1433:
1389:
1384:
1381:
1377:
1353:
1350:
1347:
1344:
1339:
1335:
1331:
1328:
1325:
1322:
1316:
1313:
1293:
1285:
1269:
1260:
1215:
1212:
1209:
1204:
1200:
1168:
1152:
1144:
1140:
1113:
1097:
1090:
1082:
1080:
1077:
1064:
1061:
1056:
1052:
1048:
1045:
1042:
1034:
1031:
1028:
1024:
1020:
1017:
1014:
1009:
1005:
1001:
998:
995:
992:
983:
980:
977:
954:
951:
946:
942:
933:
917:
914:
911:
891:
882:
866:
862:
853:
849:
833:
813:
810:
807:
798:
796:
795:zero divisors
788:
784:
776:
772:is nilpotent.
771:
767:
766:
750:
747:
744:
741:
721:
718:
715:
712:
692:
687:
681:
676:
669:
664:
658:
653:
650:
645:
640:
634:
629:
622:
617:
611:
606:
603:
595:
591:
571:
568:
566:
558:
552:
549:
543:
540:
538:
528:
520:
517:
511:
509:
502:
498:
473:
470:
467:
464:
444:
441:
438:
435:
415:
395:
375:
367:
364:
360:
356:
335:
331:
318:
314:
313:
309:
293:
290:
285:
281:
272:
254:
248:
243:
238:
231:
226:
221:
214:
209:
204:
198:
193:
190:
183:
182:
181:
180:
176:
172:
171:
167:
165:
163:
159:
154:
140:
137:
132:
128:
120:), such that
119:
115:
112:, called the
99:
92:
88:
72:
65:
49:
42:, an element
41:
34:
30:
26:
22:
2339:
2334:
2326:
2321:
2305:
2300:
2285:
2280:
2272:
2267:
2248:
2242:
2223:
2217:
2209:
2204:
2180:Reduced ring
2084:dual numbers
2081:
2061:
2053:Morse theory
2021:(again with
1923:
1848:
1764:
1628:
1528:
1261:
1086:
1078:
883:
799:
787:trivial ring
780:
770:nilsemigroup
593:
589:
177:. The matrix
155:
117:
113:
86:
37:
2366:Ring theory
2304:A. Rogers,
2271:Peirce, B.
2009:). Thus, a
1915:BRST charge
1655:Lie algebra
1167:prime ideal
317:factor ring
40:mathematics
2371:0 (number)
2360:Categories
2325:E Witten,
2196:References
2011:linear map
1972:such that
1143:nilradical
934:, because
777:Properties
408:in a ring
158:idempotent
85:is called
2212:. p. 127.
2185:Nil ideal
2175:Unipotent
2143:⊗
2129:octonions
2110:⊗
1955:∈
1817:σ
1810:±
1801:σ
1789:±
1785:σ
1738:
1668:∈
1470:∅
1461:∩
1382:−
1216:∈
1049:−
1032:−
1018:⋯
981:−
915:−
811:×
359:congruent
310:for more.
87:nilpotent
2164:See also
1284:localize
1110:form an
967:entails
428:satisfy
168:Examples
2275:. 1870.
1919:physics
1851:operand
826:matrix
592:,
315:In the
91:integer
2346:
2292:
2255:
2230:
363:modulo
353:, the
306:. See
118:degree
2005:(the
1653:be a
1446:with
1235:. So
1112:ideal
930:is a
848:field
791:0 = 1
705:Here
361:to 0
114:index
62:of a
2344:ISBN
2290:ISBN
2253:ISBN
2228:ISBN
2062:The
2051:and
1765:Any
1727:and
1629:Let
932:unit
783:unit
734:and
388:and
64:ring
2310:doi
2015:iff
1849:An
1846:.
1426:of
1262:If
884:If
854:is
800:An
38:In
2362::
2094:,
2074:.
1921:.
1757:.
1735:ad
1306::
1065:1.
881:.
797:.
596:):
572:0.
365:9.
153:.
2316:.
2312::
2261:.
2236:.
2147:O
2139:C
2114:H
2106:C
2035:2
2032:=
2029:n
1993:0
1990:=
1985:n
1981:Q
1959:N
1952:n
1932:Q
1893:0
1890:=
1885:2
1881:Q
1860:Q
1834:2
1830:/
1826:)
1821:y
1813:i
1805:x
1797:(
1794:=
1741:x
1715:]
1710:g
1705:,
1700:g
1695:[
1673:g
1665:x
1639:g
1609:I
1589:I
1585:/
1581:R
1561:R
1541:R
1512:N
1490:x
1467:=
1464:S
1456:p
1434:R
1412:p
1390:R
1385:1
1378:S
1357:}
1354:.
1351:.
1348:.
1345:,
1340:2
1336:x
1332:,
1329:x
1326:,
1323:1
1320:{
1317:=
1314:S
1294:x
1270:x
1245:N
1221:p
1213:0
1210:=
1205:n
1201:x
1178:p
1153:x
1123:N
1098:R
1062:=
1057:n
1053:x
1046:1
1043:=
1040:)
1035:1
1029:n
1025:x
1021:+
1015:+
1010:2
1006:x
1002:+
999:x
996:+
993:1
990:(
987:)
984:x
978:1
975:(
955:0
952:=
947:n
943:x
918:x
912:1
892:x
867:n
863:t
834:A
814:n
808:n
763:.
751:B
748:=
745:A
742:B
722:0
719:=
716:B
713:A
693:.
688:)
682:0
677:0
670:1
665:0
659:(
654:=
651:B
646:,
641:)
635:1
630:0
623:1
618:0
612:(
607:=
604:A
594:b
590:a
569:=
559:a
556:)
553:b
550:a
547:(
544:b
541:=
529:2
525:)
521:a
518:b
515:(
512:=
503:2
499:c
474:a
471:b
468:=
465:c
445:0
442:=
439:b
436:a
416:R
396:b
376:a
340:Z
336:9
332:/
327:Z
294:0
291:=
286:3
282:A
255:)
249:0
244:0
239:0
232:1
227:0
222:0
215:0
210:1
205:0
199:(
194:=
191:A
141:0
138:=
133:n
129:x
100:n
73:R
50:x
35:.
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