1484:
1018:
1479:{\displaystyle {\begin{aligned}\left(\sum _{i}a_{i}x^{i}\right)'&=\sum _{i}\left(a_{i}x^{i}\right)'\\&=\sum _{i}\left((a_{i})'x^{i}+a_{i}\left(x^{i}\right)'\right)\\&=\sum _{i}\left(0x^{i}+a_{i}\left(\sum _{j=1}^{i}x^{j-1}(x')x^{i-j}\right)\right)\\&=\sum _{i}\sum _{j=1}^{i}a_{i}x^{i-1}\\&=\sum _{i}ia_{i}x^{i-1}.\end{aligned}}}
2491:
will be continuously differentiable. Likewise, by choosing different classes of functions (say, the
Lipschitz class), we get different flavors of differentiability. In this way, differentiation becomes a part of algebra of functions.
437:
1646:
1781:
577:
1005:
285:
1877:
1023:
2364:
754:
929:
2250:
of scalars is commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus. The element Y–X of the ring
2015:
2212:
2153:
832:
865:
788:
486:
606:
632:
123:
509:
2489:
2469:
2449:
2429:
2409:
1013:
The formula above (i.e. the definition of the formal derivative when the coefficient ring is commutative) is a direct consequence of the aforementioned axioms:
679:
655:
460:
190:
163:
143:
88:
296:
55:. Many of the properties of the derivative are true of the formal derivative, but some, especially those that make numerical statements, are not.
2070:
is no longer a root of the resulting polynomial. The utility of this observation is that although in general not every polynomial of degree
1538:
514:
1667:
2431:, it will recapture the classical definition of the derivative. If it is carried out in the class of functions continuous in both
2227:
198:
2581:
2511:
1810:
47:. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a
2222:, so when we pass to the algebraic closure it has a multiple root that could not have been detected by factorization in
2284:
936:
696:
1010:
One may prove that this axiomatic definition yields a well-defined map respecting all of the usual ring axioms.
59:
2573:
1802:
1947:
1823:
2165:
2108:
793:
2620:
2235:
872:
933:
The map satisfies
Leibniz's law with respect to the polynomial ring's multiplication operation,
17:
465:
2591:
2526:
1880:
839:
2599:
762:
582:
8:
2521:
2391:
Actually, if the division in this definition is carried out in the class of functions of
611:
93:
48:
36:
2082:
roots counting multiplicity (this is the maximum, by the above theorem), we may pass to
491:
2474:
2454:
2434:
2414:
2394:
664:
640:
445:
175:
148:
128:
73:
52:
2090:). Once we do, we may uncover a multiple root that was not a root at all simply over
2577:
2087:
1817:
2595:
2506:
1903:
658:
1906:, and in this situation we can define multiplicity of roots; for every polynomial
1820:, because the product rule is different from saying (and it is not the case) that
432:{\displaystyle f'(x)\,=\,Df(x)=na_{n}x^{n-1}+\cdots +ia_{i}x^{i-1}+\cdots +a_{1}.}
2587:
2083:
32:
2614:
2516:
2231:
682:
2385:
2099:
1655:
166:
24:
2565:
2501:
40:
2238:(defined by polynomials with no multiple roots) and inseparable ones.
869:
The map commutes with the addition operation in the polynomial ring,
2576:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
2384:
This formulation of the derivative works equally well for a formal
693:
One may also define the formal derivative axiomatically as the map
44:
2605:
1641:{\displaystyle (r\cdot f+s\cdot g)'(x)=r\cdot f'(x)+s\cdot g'(x).}
572:{\displaystyle ir=\underbrace {r+r+\cdots +r} _{i{\text{ times}}}}
2606:
Michael
Livshits, You could simplify calculus, arXiv:0905.3611v1
2058:
is also the number of differentiations that must be performed on
1776:{\displaystyle (f\cdot g)'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x).}
2545:
2051:. It follows from the Leibniz rule that in this situation,
1894:
As in calculus, the derivative detects multiple roots. If
1889:
1497:
Formal differentiation is linear: for any two polynomials
172:
Then the formal derivative is an operation on elements of
280:{\displaystyle f(x)\,=\,a_{n}x^{n}+\cdots +a_{1}x+a_{0},}
2241:
2388:, as long as the ring of coefficients is commutative.
58:
Formal differentiation is used in algebra to test for
2477:
2471:, we get uniform differentiability, and the function
2457:
2437:
2417:
2397:
2287:
2168:
2111:
1950:
1826:
1670:
1541:
1021:
939:
875:
842:
796:
765:
699:
667:
643:
614:
585:
517:
494:
468:
448:
442:
In the above definition, for any nonnegative integer
299:
201:
178:
151:
131:
96:
76:
688:
2483:
2463:
2443:
2423:
2403:
2358:
2206:
2147:
2009:
1871:
1775:
1640:
1478:
999:
923:
859:
826:
782:
748:
673:
649:
626:
600:
571:
503:
480:
454:
431:
279:
184:
157:
137:
117:
82:
51:, which is in general impossible to define for a
2612:
2359:{\displaystyle g(X,Y)={\frac {f(Y)-f(X)}{Y-X}}.}
2226:itself. Thus, formal differentiation allows an
1879:. However, it is a homomorphism (linear map) of
1000:{\displaystyle (a\cdot b)'=a'\cdot b+a\cdot b'.}
16:For the concept in formal language theory, see
2230:notion of multiplicity. This is important in
749:{\displaystyle (\ast )^{\prime }\colon R\to R}
2377:) coincides with the formal derivative of
2254:divides Y – X for any nonnegative integer
2546:John B. Fraleigh; Victor J. Katz (2002).
2270:in one indeterminate. If the quotient in
2190:
2186:
2128:
2124:
1816:Note that the formal derivative is not a
321:
317:
218:
214:
2234:, where the distinction is made between
1890:Application to finding repeated factors
1813:for a discussion of a generalization).
169:over a single indeterminate variable.)
2613:
2010:{\displaystyle f(x)=(x-r)^{m_{r}}g(x)}
1872:{\displaystyle (f\cdot g)'=f'\cdot g'}
756:satisfying the following properties.
90:(not necessarily commutative) and let
2512:Module of relative differential forms
2242:Correspondence to analytic derivative
1926:, there exists a nonnegative integer
1811:module of relative differential forms
1793:is not commutative this is important.
2564:
2102:with three elements, the polynomial
1789:Note the order of the factors; when
1654:The formal derivative satisfies the
2369:It is then not hard to verify that
637:This definition also works even if
13:
2548:A First Course in Abstract Algebra
2162:; however, its formal derivative (
711:
14:
2632:
31:is an operation on elements of a
2207:{\displaystyle f'(x)\,=\,6x^{5}}
2148:{\displaystyle f(x)\,=\,x^{6}+1}
827:{\displaystyle r\in R\subset R.}
689:Alternative axiomatic definition
165:is not commutative, this is the
125:be the ring of polynomials over
2086:in which this is true (namely,
511:is defined as usual in a ring:
2539:
2336:
2330:
2321:
2315:
2303:
2291:
2183:
2177:
2121:
2115:
2004:
1998:
1979:
1966:
1960:
1954:
1840:
1827:
1767:
1761:
1747:
1741:
1732:
1726:
1717:
1711:
1697:
1691:
1684:
1671:
1632:
1626:
1606:
1600:
1580:
1574:
1567:
1542:
1324:
1313:
1162:
1148:
953:
940:
889:
876:
818:
812:
743:
737:
731:
728:
722:
707:
700:
334:
328:
314:
308:
290:then its formal derivative is
211:
205:
112:
106:
60:multiple roots of a polynomial
1:
2574:Graduate Texts in Mathematics
2532:
1488:
924:{\displaystyle (a+b)'=a'+b'.}
65:
39:that mimics the form of the
7:
2495:
10:
2637:
2558:
2236:separable field extensions
1797:These two properties make
15:
2381:as it was defined above.
2214:) is zero since 3 = 0 in
1493:It can be verified that:
836:The normalization axiom,
2258:, and therefore divides
2218:and in any extension of
2550:. Pearson. p. 443.
2266:(X) for any polynomial
2043:is the multiplicity of
659:multiplicative identity
2485:
2465:
2445:
2425:
2405:
2360:
2208:
2149:
2011:
1886:, by the above rules.
1873:
1777:
1642:
1480:
1393:
1296:
1001:
925:
861:
828:
784:
750:
675:
651:
628:
602:
573:
505:
482:
481:{\displaystyle r\in R}
456:
433:
281:
186:
159:
139:
119:
84:
2486:
2466:
2446:
2426:
2406:
2361:
2209:
2150:
2012:
1874:
1778:
1643:
1481:
1373:
1276:
1002:
926:
862:
860:{\displaystyle x'=1.}
829:
785:
751:
676:
652:
629:
603:
574:
506:
483:
457:
434:
282:
187:
160:
140:
120:
85:
18:Brzozowski derivative
2527:Pincherle derivative
2475:
2455:
2435:
2415:
2395:
2285:
2166:
2109:
1948:
1824:
1668:
1539:
1019:
937:
873:
840:
794:
783:{\displaystyle r'=0}
763:
697:
665:
641:
612:
601:{\displaystyle ir=0}
583:
515:
492:
466:
446:
297:
199:
176:
149:
129:
94:
74:
2522:Formal power series
2094:. For example, if
627:{\displaystyle i=0}
118:{\displaystyle A=R}
37:formal power series
2481:
2461:
2441:
2421:
2401:
2356:
2204:
2145:
2088:algebraic closures
2007:
1918:and every element
1869:
1773:
1638:
1476:
1474:
1439:
1372:
1239:
1142:
1087:
1041:
997:
921:
857:
824:
780:
746:
671:
647:
624:
598:
569:
568:
556:
504:{\displaystyle ir}
501:
478:
452:
429:
277:
182:
155:
135:
115:
80:
2583:978-0-387-95385-4
2484:{\displaystyle f}
2464:{\displaystyle Y}
2444:{\displaystyle X}
2424:{\displaystyle X}
2404:{\displaystyle Y}
2351:
1933:and a polynomial
1818:ring homomorphism
1430:
1363:
1230:
1133:
1078:
1032:
674:{\displaystyle R}
650:{\displaystyle R}
565:
529:
527:
455:{\displaystyle i}
185:{\displaystyle A}
158:{\displaystyle R}
138:{\displaystyle R}
83:{\displaystyle R}
29:formal derivative
2628:
2621:Abstract algebra
2602:
2552:
2551:
2543:
2507:Euclidean domain
2490:
2488:
2487:
2482:
2470:
2468:
2467:
2462:
2450:
2448:
2447:
2442:
2430:
2428:
2427:
2422:
2410:
2408:
2407:
2402:
2365:
2363:
2362:
2357:
2352:
2350:
2339:
2310:
2213:
2211:
2210:
2205:
2203:
2202:
2176:
2158:has no roots in
2154:
2152:
2151:
2146:
2138:
2137:
2084:field extensions
2035:
2031:
2016:
2014:
2013:
2008:
1994:
1993:
1992:
1991:
1904:Euclidean domain
1898:is a field then
1878:
1876:
1875:
1870:
1868:
1857:
1846:
1782:
1780:
1779:
1774:
1760:
1710:
1690:
1647:
1645:
1644:
1639:
1625:
1599:
1573:
1485:
1483:
1482:
1477:
1475:
1468:
1467:
1452:
1451:
1438:
1423:
1419:
1418:
1403:
1402:
1392:
1387:
1371:
1356:
1352:
1348:
1347:
1343:
1342:
1341:
1323:
1312:
1311:
1295:
1290:
1270:
1269:
1257:
1256:
1238:
1223:
1219:
1215:
1214:
1210:
1206:
1205:
1191:
1190:
1178:
1177:
1168:
1160:
1159:
1141:
1126:
1122:
1118:
1114:
1113:
1112:
1103:
1102:
1086:
1070:
1066:
1062:
1061:
1060:
1051:
1050:
1040:
1006:
1004:
1003:
998:
993:
970:
959:
930:
928:
927:
922:
917:
906:
895:
866:
864:
863:
858:
850:
833:
831:
830:
825:
789:
787:
786:
781:
773:
755:
753:
752:
747:
715:
714:
680:
678:
677:
672:
657:does not have a
656:
654:
653:
648:
633:
631:
630:
625:
607:
605:
604:
599:
578:
576:
575:
570:
567:
566:
563:
557:
552:
510:
508:
507:
502:
487:
485:
484:
479:
461:
459:
458:
453:
438:
436:
435:
430:
425:
424:
406:
405:
390:
389:
368:
367:
352:
351:
307:
286:
284:
283:
278:
273:
272:
257:
256:
238:
237:
228:
227:
191:
189:
188:
183:
164:
162:
161:
156:
144:
142:
141:
136:
124:
122:
121:
116:
89:
87:
86:
81:
2636:
2635:
2631:
2630:
2629:
2627:
2626:
2625:
2611:
2610:
2584:
2561:
2556:
2555:
2544:
2540:
2535:
2498:
2476:
2473:
2472:
2456:
2453:
2452:
2436:
2433:
2432:
2416:
2413:
2412:
2396:
2393:
2392:
2340:
2311:
2309:
2286:
2283:
2282:
2244:
2198:
2194:
2169:
2167:
2164:
2163:
2133:
2129:
2110:
2107:
2106:
2056:
2041:
2033:
2029:
1987:
1983:
1982:
1978:
1949:
1946:
1945:
1931:
1892:
1861:
1850:
1839:
1825:
1822:
1821:
1753:
1703:
1683:
1669:
1666:
1665:
1618:
1592:
1566:
1540:
1537:
1536:
1491:
1473:
1472:
1457:
1453:
1447:
1443:
1434:
1421:
1420:
1408:
1404:
1398:
1394:
1388:
1377:
1367:
1354:
1353:
1331:
1327:
1316:
1301:
1297:
1291:
1280:
1275:
1271:
1265:
1261:
1252:
1248:
1244:
1240:
1234:
1221:
1220:
1201:
1197:
1193:
1192:
1186:
1182:
1173:
1169:
1161:
1155:
1151:
1147:
1143:
1137:
1124:
1123:
1108:
1104:
1098:
1094:
1093:
1089:
1088:
1082:
1071:
1056:
1052:
1046:
1042:
1036:
1031:
1027:
1026:
1022:
1020:
1017:
1016:
986:
963:
952:
938:
935:
934:
910:
899:
888:
874:
871:
870:
843:
841:
838:
837:
795:
792:
791:
766:
764:
761:
760:
710:
706:
698:
695:
694:
691:
666:
663:
662:
642:
639:
638:
613:
610:
609:
584:
581:
580:
562:
558:
530:
528:
516:
513:
512:
493:
490:
489:
467:
464:
463:
447:
444:
443:
420:
416:
395:
391:
385:
381:
357:
353:
347:
343:
300:
298:
295:
294:
268:
264:
252:
248:
233:
229:
223:
219:
200:
197:
196:
177:
174:
173:
150:
147:
146:
130:
127:
126:
95:
92:
91:
75:
72:
71:
68:
33:polynomial ring
21:
12:
11:
5:
2634:
2624:
2623:
2609:
2608:
2603:
2582:
2560:
2557:
2554:
2553:
2537:
2536:
2534:
2531:
2530:
2529:
2524:
2519:
2514:
2509:
2504:
2497:
2494:
2480:
2460:
2440:
2420:
2411:continuous at
2400:
2367:
2366:
2355:
2349:
2346:
2343:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2274:is denoted by
2246:When the ring
2243:
2240:
2201:
2197:
2193:
2189:
2185:
2182:
2179:
2175:
2172:
2156:
2155:
2144:
2141:
2136:
2132:
2127:
2123:
2120:
2117:
2114:
2054:
2039:
2018:
2017:
2006:
2003:
2000:
1997:
1990:
1986:
1981:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1929:
1891:
1888:
1867:
1864:
1860:
1856:
1853:
1849:
1845:
1842:
1838:
1835:
1832:
1829:
1795:
1794:
1786:
1785:
1784:
1783:
1772:
1769:
1766:
1763:
1759:
1756:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1709:
1706:
1702:
1699:
1696:
1693:
1689:
1686:
1682:
1679:
1676:
1673:
1660:
1659:
1651:
1650:
1649:
1648:
1637:
1634:
1631:
1628:
1624:
1621:
1617:
1614:
1611:
1608:
1605:
1602:
1598:
1595:
1591:
1588:
1585:
1582:
1579:
1576:
1572:
1569:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1531:
1530:
1490:
1487:
1471:
1466:
1463:
1460:
1456:
1450:
1446:
1442:
1437:
1433:
1429:
1426:
1424:
1422:
1417:
1414:
1411:
1407:
1401:
1397:
1391:
1386:
1383:
1380:
1376:
1370:
1366:
1362:
1359:
1357:
1355:
1351:
1346:
1340:
1337:
1334:
1330:
1326:
1322:
1319:
1315:
1310:
1307:
1304:
1300:
1294:
1289:
1286:
1283:
1279:
1274:
1268:
1264:
1260:
1255:
1251:
1247:
1243:
1237:
1233:
1229:
1226:
1224:
1222:
1218:
1213:
1209:
1204:
1200:
1196:
1189:
1185:
1181:
1176:
1172:
1167:
1164:
1158:
1154:
1150:
1146:
1140:
1136:
1132:
1129:
1127:
1125:
1121:
1117:
1111:
1107:
1101:
1097:
1092:
1085:
1081:
1077:
1074:
1072:
1069:
1065:
1059:
1055:
1049:
1045:
1039:
1035:
1030:
1025:
1024:
1008:
1007:
996:
992:
989:
985:
982:
979:
976:
973:
969:
966:
962:
958:
955:
951:
948:
945:
942:
931:
920:
916:
913:
909:
905:
902:
898:
894:
891:
887:
884:
881:
878:
867:
856:
853:
849:
846:
834:
823:
820:
817:
814:
811:
808:
805:
802:
799:
779:
776:
772:
769:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
713:
709:
705:
702:
690:
687:
670:
646:
623:
620:
617:
597:
594:
591:
588:
561:
555:
551:
548:
545:
542:
539:
536:
533:
526:
523:
520:
500:
497:
477:
474:
471:
451:
440:
439:
428:
423:
419:
415:
412:
409:
404:
401:
398:
394:
388:
384:
380:
377:
374:
371:
366:
363:
360:
356:
350:
346:
342:
339:
336:
333:
330:
327:
324:
320:
316:
313:
310:
306:
303:
288:
287:
276:
271:
267:
263:
260:
255:
251:
247:
244:
241:
236:
232:
226:
222:
217:
213:
210:
207:
204:
181:
154:
134:
114:
111:
108:
105:
102:
99:
79:
67:
64:
9:
6:
4:
3:
2:
2633:
2622:
2619:
2618:
2616:
2607:
2604:
2601:
2597:
2593:
2589:
2585:
2579:
2575:
2571:
2567:
2563:
2562:
2549:
2542:
2538:
2528:
2525:
2523:
2520:
2518:
2517:Galois theory
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2499:
2493:
2478:
2458:
2438:
2418:
2398:
2389:
2387:
2382:
2380:
2376:
2372:
2353:
2347:
2344:
2341:
2333:
2327:
2324:
2318:
2312:
2306:
2300:
2297:
2294:
2288:
2281:
2280:
2279:
2277:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2239:
2237:
2233:
2232:Galois theory
2229:
2225:
2221:
2217:
2199:
2195:
2191:
2187:
2180:
2173:
2170:
2161:
2142:
2139:
2134:
2130:
2125:
2118:
2112:
2105:
2104:
2103:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2050:
2047:as a root of
2046:
2042:
2027:
2023:
2001:
1995:
1988:
1984:
1975:
1972:
1969:
1963:
1957:
1951:
1944:
1943:
1942:
1940:
1936:
1932:
1925:
1921:
1917:
1913:
1909:
1905:
1901:
1897:
1887:
1885:
1883:
1865:
1862:
1858:
1854:
1851:
1847:
1843:
1836:
1833:
1830:
1819:
1814:
1812:
1808:
1804:
1800:
1792:
1788:
1787:
1770:
1764:
1757:
1754:
1750:
1744:
1738:
1735:
1729:
1723:
1720:
1714:
1707:
1704:
1700:
1694:
1687:
1680:
1677:
1674:
1664:
1663:
1662:
1661:
1657:
1653:
1652:
1635:
1629:
1622:
1619:
1615:
1612:
1609:
1603:
1596:
1593:
1589:
1586:
1583:
1577:
1570:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1535:
1534:
1533:
1532:
1528:
1524:
1520:
1517:and elements
1516:
1512:
1508:
1504:
1500:
1496:
1495:
1494:
1486:
1469:
1464:
1461:
1458:
1454:
1448:
1444:
1440:
1435:
1431:
1427:
1425:
1415:
1412:
1409:
1405:
1399:
1395:
1389:
1384:
1381:
1378:
1374:
1368:
1364:
1360:
1358:
1349:
1344:
1338:
1335:
1332:
1328:
1320:
1317:
1308:
1305:
1302:
1298:
1292:
1287:
1284:
1281:
1277:
1272:
1266:
1262:
1258:
1253:
1249:
1245:
1241:
1235:
1231:
1227:
1225:
1216:
1211:
1207:
1202:
1198:
1194:
1187:
1183:
1179:
1174:
1170:
1165:
1156:
1152:
1144:
1138:
1134:
1130:
1128:
1119:
1115:
1109:
1105:
1099:
1095:
1090:
1083:
1079:
1075:
1073:
1067:
1063:
1057:
1053:
1047:
1043:
1037:
1033:
1028:
1014:
1011:
994:
990:
987:
983:
980:
977:
974:
971:
967:
964:
960:
956:
949:
946:
943:
932:
918:
914:
911:
907:
903:
900:
896:
892:
885:
882:
879:
868:
854:
851:
847:
844:
835:
821:
815:
809:
806:
803:
800:
797:
777:
774:
770:
767:
759:
758:
757:
740:
734:
725:
719:
716:
703:
686:
684:
668:
660:
644:
635:
621:
618:
615:
595:
592:
589:
586:
559:
553:
549:
546:
543:
540:
537:
534:
531:
524:
521:
518:
498:
495:
475:
472:
469:
449:
426:
421:
417:
413:
410:
407:
402:
399:
396:
392:
386:
382:
378:
375:
372:
369:
364:
361:
358:
354:
348:
344:
340:
337:
331:
325:
322:
318:
311:
304:
301:
293:
292:
291:
274:
269:
265:
261:
258:
253:
249:
245:
242:
239:
234:
230:
224:
220:
215:
208:
202:
195:
194:
193:
179:
170:
168:
152:
132:
109:
103:
100:
97:
77:
63:
61:
56:
54:
50:
46:
42:
38:
35:or a ring of
34:
30:
26:
19:
2569:
2547:
2541:
2390:
2386:power series
2383:
2378:
2374:
2370:
2368:
2275:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2245:
2223:
2219:
2215:
2159:
2157:
2100:finite field
2095:
2091:
2079:
2075:
2071:
2067:
2063:
2059:
2052:
2048:
2044:
2037:
2025:
2021:
2019:
1941:) such that
1938:
1934:
1927:
1923:
1919:
1915:
1911:
1907:
1899:
1895:
1893:
1881:
1815:
1806:
1798:
1796:
1790:
1656:product rule
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1492:
1015:
1012:
1009:
692:
636:
441:
289:
171:
167:free algebra
69:
57:
28:
22:
2566:Lang, Serge
564: times
192:, where if
70:Fix a ring
25:mathematics
2600:0984.00001
2533:References
2502:Derivative
2373:(X,X) (in
1803:derivation
1489:Properties
661:(that is,
66:Definition
41:derivative
2345:−
2325:−
2228:effective
2066:) before
1973:−
1859:⋅
1834:⋅
1751:⋅
1721:⋅
1678:⋅
1616:⋅
1590:⋅
1561:⋅
1549:⋅
1462:−
1432:∑
1413:−
1375:∑
1365:∑
1336:−
1306:−
1278:∑
1232:∑
1135:∑
1080:∑
1034:∑
984:⋅
972:⋅
947:⋅
807:⊂
801:∈
732:→
717::
712:′
704:∗
554:⏟
544:⋯
473:∈
411:⋯
400:−
373:⋯
362:−
243:⋯
2615:Category
2568:(2002),
2496:See also
2174:′
1884:-modules
1866:′
1855:′
1844:′
1758:′
1708:′
1688:′
1623:′
1597:′
1571:′
1321:′
1212:′
1166:′
1120:′
1068:′
991:′
968:′
957:′
915:′
904:′
893:′
848:′
790:for all
771:′
305:′
45:calculus
2592:1878556
2570:Algebra
2559:Sources
2278:, then
2098:is the
2034:
2030:
1529:we have
2598:
2590:
2580:
2262:(Y) –
2020:where
579:(with
145:. (If
27:, the
1914:) in
1902:is a
1809:(see
1513:) in
681:is a
49:limit
43:from
2578:ISBN
2451:and
2078:has
2036:0.
462:and
53:ring
2596:Zbl
2074:in
1922:of
1805:on
1525:of
685:).
683:rng
634:).
608:if
23:In
2617::
2594:,
2588:MR
2586:,
2572:,
1801:a
1505:),
855:1.
488:,
62:.
2479:f
2459:Y
2439:X
2419:X
2399:Y
2379:f
2375:R
2371:g
2354:.
2348:X
2342:Y
2337:)
2334:X
2331:(
2328:f
2322:)
2319:Y
2316:(
2313:f
2307:=
2304:)
2301:Y
2298:,
2295:X
2292:(
2289:g
2276:g
2272:R
2268:f
2264:f
2260:f
2256:n
2252:R
2248:R
2224:R
2220:R
2216:R
2200:5
2196:x
2192:6
2188:=
2184:)
2181:x
2178:(
2171:f
2160:R
2143:1
2140:+
2135:6
2131:x
2126:=
2122:)
2119:x
2116:(
2113:f
2096:R
2092:R
2080:n
2076:R
2072:n
2068:r
2064:x
2062:(
2060:f
2055:r
2053:m
2049:f
2045:r
2040:r
2038:m
2032:≠
2028:)
2026:r
2024:(
2022:g
2005:)
2002:x
1999:(
1996:g
1989:r
1985:m
1980:)
1976:r
1970:x
1967:(
1964:=
1961:)
1958:x
1955:(
1952:f
1939:x
1937:(
1935:g
1930:r
1928:m
1924:R
1920:r
1916:R
1912:x
1910:(
1908:f
1900:R
1896:R
1882:R
1863:g
1852:f
1848:=
1841:)
1837:g
1831:f
1828:(
1807:A
1799:D
1791:R
1771:.
1768:)
1765:x
1762:(
1755:g
1748:)
1745:x
1742:(
1739:f
1736:+
1733:)
1730:x
1727:(
1724:g
1718:)
1715:x
1712:(
1705:f
1701:=
1698:)
1695:x
1692:(
1685:)
1681:g
1675:f
1672:(
1658::
1636:.
1633:)
1630:x
1627:(
1620:g
1613:s
1610:+
1607:)
1604:x
1601:(
1594:f
1587:r
1584:=
1581:)
1578:x
1575:(
1568:)
1564:g
1558:s
1555:+
1552:f
1546:r
1543:(
1527:R
1523:s
1521:,
1519:r
1515:R
1511:x
1509:(
1507:g
1503:x
1501:(
1499:f
1470:.
1465:1
1459:i
1455:x
1449:i
1445:a
1441:i
1436:i
1428:=
1416:1
1410:i
1406:x
1400:i
1396:a
1390:i
1385:1
1382:=
1379:j
1369:i
1361:=
1350:)
1345:)
1339:j
1333:i
1329:x
1325:)
1318:x
1314:(
1309:1
1303:j
1299:x
1293:i
1288:1
1285:=
1282:j
1273:(
1267:i
1263:a
1259:+
1254:i
1250:x
1246:0
1242:(
1236:i
1228:=
1217:)
1208:)
1203:i
1199:x
1195:(
1188:i
1184:a
1180:+
1175:i
1171:x
1163:)
1157:i
1153:a
1149:(
1145:(
1139:i
1131:=
1116:)
1110:i
1106:x
1100:i
1096:a
1091:(
1084:i
1076:=
1064:)
1058:i
1054:x
1048:i
1044:a
1038:i
1029:(
995:.
988:b
981:a
978:+
975:b
965:a
961:=
954:)
950:b
944:a
941:(
919:.
912:b
908:+
901:a
897:=
890:)
886:b
883:+
880:a
877:(
852:=
845:x
822:.
819:]
816:x
813:[
810:R
804:R
798:r
778:0
775:=
768:r
744:]
741:x
738:[
735:R
729:]
726:x
723:[
720:R
708:)
701:(
669:R
645:R
622:0
619:=
616:i
596:0
593:=
590:r
587:i
560:i
550:r
547:+
541:+
538:r
535:+
532:r
525:=
522:r
519:i
499:r
496:i
476:R
470:r
450:i
427:.
422:1
418:a
414:+
408:+
403:1
397:i
393:x
387:i
383:a
379:i
376:+
370:+
365:1
359:n
355:x
349:n
345:a
341:n
338:=
335:)
332:x
329:(
326:f
323:D
319:=
315:)
312:x
309:(
302:f
275:,
270:0
266:a
262:+
259:x
254:1
250:a
246:+
240:+
235:n
231:x
225:n
221:a
216:=
212:)
209:x
206:(
203:f
180:A
153:R
133:R
113:]
110:x
107:[
104:R
101:=
98:A
78:R
20:.
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