1663:
1210:
66:
1160:
In a general ring, an ideal may not be invertible (in fact, already the definition of a fractional ideal is not clear). However, over a
Noetherian integral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki's
1587:
1977:
102:
Ideals in a finitely generated algebra over a field (that is, a quotient of a polynomial ring over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if
756:
1433:
1856:
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280:
1803:
983:
906:
2017:
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397:
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954:
928:
838:
1872:
1143:
539:
1079:
309:
1336:
1309:
426:
346:
1405:
1149:. In other words, fractional ideals form a group under a product. The quotient of the group of fractional ideals by the subgroup of principal ideals is then the
2061:
2041:
1774:
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366:
193:
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1693:
Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.
2147:
2101:
46:, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.)
671:
2121:
2189:
2133:
1582:{\displaystyle (I:J^{\infty })=\{f\in A\mid fJ^{n}\subset I,n\gg 0\}=\bigcup _{n>0}\operatorname {Ann} _{A}((J^{n}+I)/I)}
2184:
1008:(e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve) with the field of fractions
2194:
1245:
931:
172:
1808:
2139:
97:
589:
986:
20:
1163:
243:
1779:
959:
847:
2072:
1986:
1972:{\displaystyle \Gamma _{I}(M):=\Gamma (Y,{\widetilde {M}})=\varinjlim \operatorname {Hom} (I^{n},M)}
934:. The resulting complete metric space has a structure of a ring that extended the ring structure of
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43:
35:
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and is a closure operation (this notion is closely related to the study of local cohomology).
1240:
There are several operations on ideals that play roles of closures. The most basic one is the
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1049:
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39:
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1002:
402:
322:
204:
19:
This article is about the mathematical theory. For the usage in political philosophy, see
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2109:
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2046:
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130:
106:
1662:
1209:
65:
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2129:
1636:
168:
2161:
1175:
841:
1193:, it is more convenient to use a generalization of an ideal class group called an
2138:, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK:
49:
Throughout the articles, rings refer to commutative rings. See also the article
27:
1311:'s whose radicals are minimal (don’t contain any of the radicals of other
147:
is the intersection of all maximal ideals containing the ideal (because
751:{\displaystyle B(x,r)=\{z\in \mathbb {Z} \mid |z-x|_{p}<r\}}
1657:
1204:
1145:, where the product on the left is a product of submodules of
60:
2116:, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995,
1186:(often the two are the same; e.g., for Dedekind domains).
2114:
Commutative
Algebra with a View Toward Algebraic Geometry
53:
for basic operations such as sum or products of ideals.
1674:
1358:; this intersection is then called the unmixed part of
1221:
123:
is a finitely generated algebra over a field, then the
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109:
1174:, when it can be defined, is closely related to the
16:
Theory of ideals in commutative rings in mathematics
57:
Ideals in a finitely generated algebra over a field
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2011:
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139:
115:
1851:{\displaystyle Y=\operatorname {Spec} (R)-V(I)}
42:. While the notion of an ideal exists also for
2135:Integral closure of ideals, rings, and modules
1248:. Given an irredundant primary decomposition
171:). This may be thought of as an extension of
8:
1508:
1462:
745:
696:
1048:is invertible in the sense: there exists a
1189:In algebraic number theory, especially in
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2028:
1994:
1988:
1954:
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1911:
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673:
631:
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510:
486:
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404:
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353:
324:
290:
257:
245:
180:
152:
132:
108:
658:{\displaystyle x+p^{n}A=B(x,p^{-(n-1)})}
1866:). Unwinding the definition, one sees:
428:an ideal generated by a prime number
315:-adic topology. It is also called an
217:, then it determines the topology on
7:
2094:Introduction to Commutative Algebra
1991:
1898:
1877:
1451:
1378:. It is also a closure operation.
14:
275:{\displaystyle x+I^{n}\subset U.}
1798:{\displaystyle {\widetilde {M}}}
1661:
1654:Local cohomology in ideal theory
1208:
978:{\displaystyle \mathbb {Z} _{p}}
901:{\displaystyle d(x,y)=|x-y|_{p}}
64:
758:denotes an open ball of radius
199:Topology determined by an ideal
175:, which concerns the case when
2012:{\displaystyle \Gamma _{I}(M)}
2006:
2000:
1966:
1947:
1922:
1901:
1892:
1886:
1845:
1839:
1830:
1824:
1576:
1565:
1546:
1543:
1456:
1437:
1338:'s) is uniquely determined by
888:
873:
866:
854:
729:
714:
690:
678:
652:
647:
635:
618:
498:{\displaystyle |x|_{p}=p^{-n}}
469:
460:
392:{\displaystyle A=\mathbb {Z} }
311:. This topology is called the
1:
1592:is called the saturation of
1277:{\displaystyle I=\cap Q_{i}}
1246:integral closure of an ideal
949:{\displaystyle \mathbb {Z} }
923:{\displaystyle \mathbb {Z} }
833:{\displaystyle \mathbb {Z} }
1862:of the sheaf associated to
1138:{\displaystyle I\,I^{-1}=A}
399:, the ring of integers and
348:is generated by an element
2211:
2140:Cambridge University Press
1646:
956:; this ring is denoted as
202:
98:finitely generated algebra
95:
18:
1170:The ideal class group of
173:Hilbert's Nullstellensatz
1716:be a module over a ring
534:{\displaystyle x=p^{n}y}
2086:Atiyah, Michael Francis
21:Ideal theory (politics)
2190:History of mathematics
2057:
2037:
2013:
1973:
1852:
1799:
1770:
1750:
1730:
1710:
1626:
1606:
1583:
1421:
1401:
1372:
1352:
1332:
1305:
1284:, the intersection of
1278:
1139:
1099:
1075:
1074:{\displaystyle I^{-1}}
1042:
1022:
979:
950:
924:
902:
834:
812:
792:
772:
752:
659:
577:
555:
535:
499:
446:
422:
393:
362:
342:
305:
304:{\displaystyle n>0}
276:
213:is an ideal in a ring
195:is a polynomial ring.
189:
161:
141:
117:
2058:
2038:
2014:
1974:
1853:
1800:
1776:determines the sheaf
1771:
1751:
1731:
1711:
1627:
1607:
1584:
1422:
1402:
1373:
1353:
1333:
1331:{\displaystyle Q_{j}}
1306:
1304:{\displaystyle Q_{i}}
1279:
1167:gives such a theory.
1140:
1100:
1076:
1043:
1023:
980:
951:
925:
908:. As a metric space,
903:
835:
813:
793:
773:
753:
660:
578:
556:
536:
500:
447:
423:
394:
363:
343:
306:
277:
229:is open if, for each
190:
162:
142:
118:
44:non-commutative rings
2185:Ideals (ring theory)
2073:System of parameters
2047:
2027:
1987:
1873:
1858:(the restriction to
1809:
1780:
1760:
1740:
1720:
1700:
1616:
1596:
1434:
1411:
1385:
1362:
1342:
1315:
1288:
1252:
1109:
1089:
1055:
1032:
1012:
960:
938:
912:
848:
822:
802:
782:
762:
672:
590:
567:
545:
509:
456:
436:
421:{\displaystyle I=pA}
403:
375:
352:
341:{\displaystyle I=aA}
323:
289:
244:
179:
151:
131:
107:
2195:Commutative algebra
1400:{\displaystyle I,J}
1242:radical of an ideal
1164:Algèbre commutative
840:is the same as the
432:. For each integer
125:radical of an ideal
51:ideal (ring theory)
2096:, Westview Press,
2053:
2033:
2009:
1969:
1936:
1848:
1795:
1766:
1746:
1726:
1706:
1673:. You can help by
1622:
1602:
1579:
1529:
1417:
1397:
1368:
1348:
1328:
1301:
1274:
1220:. You can help by
1201:Closure operations
1191:class field theory
1135:
1095:
1071:
1038:
1018:
985:and is called the
975:
946:
920:
898:
844:topology given by
830:
818:-adic topology on
808:
788:
768:
748:
655:
573:
551:
531:
495:
442:
418:
389:
371:For example, take
358:
338:
319:-adic topology if
301:
272:
185:
157:
137:
113:
76:. You can help by
2149:978-0-521-68860-4
2103:978-0-201-40751-8
2056:{\displaystyle I}
2036:{\displaystyle M}
1929:
1919:
1792:
1769:{\displaystyle M}
1749:{\displaystyle I}
1729:{\displaystyle R}
1709:{\displaystyle M}
1691:
1690:
1625:{\displaystyle J}
1605:{\displaystyle I}
1514:
1420:{\displaystyle A}
1371:{\displaystyle I}
1351:{\displaystyle I}
1244:. Another is the
1238:
1237:
1195:idele class group
1151:ideal class group
1098:{\displaystyle K}
1041:{\displaystyle I}
1021:{\displaystyle K}
997:Ideal class group
811:{\displaystyle p}
791:{\displaystyle x}
771:{\displaystyle r}
583:. Then, clearly,
576:{\displaystyle p}
554:{\displaystyle y}
445:{\displaystyle x}
361:{\displaystyle a}
285:for some integer
188:{\displaystyle A}
160:{\displaystyle A}
140:{\displaystyle A}
116:{\displaystyle A}
94:
93:
40:commutative rings
34:is the theory of
2202:
2171:
2170:
2169:
2160:, archived from
2106:
2062:
2060:
2059:
2054:
2043:with respect to
2042:
2040:
2039:
2034:
2018:
2016:
2015:
2010:
1999:
1998:
1978:
1976:
1975:
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1959:
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1937:
1921:
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1912:
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1884:
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1855:
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1804:
1802:
1801:
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1733:
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1727:
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1712:
1707:
1686:
1683:
1665:
1658:
1643:Reduction theory
1631:
1629:
1628:
1623:
1612:with respect to
1611:
1609:
1608:
1603:
1588:
1586:
1585:
1580:
1572:
1558:
1557:
1539:
1538:
1528:
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1454:
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1406:
1404:
1403:
1398:
1377:
1375:
1374:
1369:
1357:
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1349:
1337:
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1334:
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1212:
1205:
1144:
1142:
1141:
1136:
1128:
1127:
1104:
1102:
1101:
1096:
1080:
1078:
1077:
1072:
1070:
1069:
1050:fractional ideal
1047:
1045:
1044:
1039:
1027:
1025:
1024:
1019:
984:
982:
981:
976:
974:
973:
968:
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953:
952:
947:
945:
929:
927:
926:
921:
919:
907:
905:
904:
899:
897:
896:
891:
876:
839:
837:
836:
831:
829:
817:
815:
814:
809:
797:
795:
794:
789:
777:
775:
774:
769:
757:
755:
754:
749:
738:
737:
732:
717:
709:
664:
662:
661:
656:
651:
650:
608:
607:
582:
580:
579:
574:
560:
558:
557:
552:
540:
538:
537:
532:
527:
526:
504:
502:
501:
496:
494:
493:
478:
477:
472:
463:
451:
449:
448:
443:
427:
425:
424:
419:
398:
396:
395:
390:
388:
367:
365:
364:
359:
347:
345:
344:
339:
310:
308:
307:
302:
281:
279:
278:
273:
262:
261:
194:
192:
191:
186:
166:
164:
163:
158:
146:
144:
143:
138:
122:
120:
119:
114:
89:
86:
68:
61:
2210:
2209:
2205:
2204:
2203:
2201:
2200:
2199:
2175:
2174:
2167:
2165:
2150:
2128:Huneke, Craig;
2127:
2110:Eisenbud, David
2104:
2090:Macdonald, I.G.
2084:
2081:
2069:
2045:
2044:
2025:
2024:
2021:ideal transform
1990:
1985:
1984:
1950:
1876:
1871:
1870:
1807:
1806:
1778:
1777:
1758:
1757:
1756:an ideal. Then
1738:
1737:
1718:
1717:
1698:
1697:
1687:
1681:
1678:
1671:needs expansion
1656:
1651:
1649:Ideal reduction
1645:
1614:
1613:
1594:
1593:
1549:
1530:
1480:
1446:
1432:
1431:
1409:
1408:
1383:
1382:
1360:
1359:
1340:
1339:
1318:
1313:
1312:
1291:
1286:
1285:
1264:
1250:
1249:
1234:
1228:
1225:
1218:needs expansion
1203:
1116:
1107:
1106:
1087:
1086:
1058:
1053:
1052:
1030:
1029:
1010:
1009:
1003:Dedekind domain
999:
963:
958:
957:
936:
935:
910:
909:
886:
846:
845:
820:
819:
800:
799:
780:
779:
760:
759:
727:
670:
669:
627:
599:
588:
587:
565:
564:
543:
542:
518:
507:
506:
482:
467:
454:
453:
434:
433:
401:
400:
373:
372:
350:
349:
321:
320:
287:
286:
253:
242:
241:
221:where a subset
207:
205:I-adic topology
201:
177:
176:
149:
148:
129:
128:
105:
104:
100:
90:
84:
81:
74:needs expansion
59:
24:
17:
12:
11:
5:
2208:
2206:
2198:
2197:
2192:
2187:
2177:
2176:
2173:
2172:
2148:
2130:Swanson, Irena
2125:
2107:
2102:
2080:
2077:
2076:
2075:
2068:
2065:
2052:
2032:
2019:is called the
2008:
2005:
2002:
1997:
1993:
1981:
1980:
1968:
1965:
1962:
1957:
1953:
1949:
1946:
1943:
1940:
1935:
1932:
1927:
1924:
1918:
1915:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1883:
1879:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1791:
1788:
1765:
1745:
1725:
1705:
1689:
1688:
1668:
1666:
1655:
1652:
1647:Main article:
1644:
1641:
1621:
1601:
1590:
1589:
1578:
1575:
1571:
1567:
1564:
1561:
1556:
1552:
1548:
1545:
1542:
1537:
1533:
1527:
1524:
1521:
1517:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1487:
1483:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1453:
1449:
1445:
1442:
1439:
1416:
1396:
1393:
1390:
1367:
1347:
1325:
1321:
1298:
1294:
1271:
1267:
1263:
1260:
1257:
1236:
1235:
1215:
1213:
1202:
1199:
1134:
1131:
1126:
1123:
1119:
1114:
1094:
1085:-submodule of
1068:
1065:
1061:
1037:
1017:
998:
995:
991:-adic integers
972:
967:
944:
918:
895:
890:
885:
882:
879:
875:
871:
868:
865:
862:
859:
856:
853:
828:
807:
787:
767:
747:
744:
741:
736:
731:
726:
723:
720:
716:
712:
708:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
666:
665:
654:
649:
646:
643:
640:
637:
634:
630:
626:
623:
620:
617:
614:
611:
606:
602:
598:
595:
572:
550:
530:
525:
521:
517:
514:
492:
489:
485:
481:
476:
471:
466:
462:
441:
417:
414:
411:
408:
387:
383:
380:
357:
337:
334:
331:
328:
300:
297:
294:
283:
282:
271:
268:
265:
260:
256:
252:
249:
203:Main article:
200:
197:
184:
156:
136:
112:
92:
91:
71:
69:
58:
55:
15:
13:
10:
9:
6:
4:
3:
2:
2207:
2196:
2193:
2191:
2188:
2186:
2183:
2182:
2180:
2164:on 2019-11-15
2163:
2159:
2155:
2151:
2145:
2141:
2137:
2136:
2131:
2126:
2123:
2122:0-387-94268-8
2119:
2115:
2111:
2108:
2105:
2099:
2095:
2091:
2087:
2083:
2082:
2078:
2074:
2071:
2070:
2066:
2064:
2050:
2030:
2022:
2003:
1995:
1963:
1960:
1955:
1951:
1944:
1941:
1938:
1933:
1930:
1925:
1916:
1913:
1907:
1904:
1895:
1889:
1881:
1869:
1868:
1867:
1865:
1861:
1842:
1836:
1833:
1827:
1821:
1818:
1815:
1812:
1789:
1786:
1763:
1743:
1723:
1703:
1694:
1685:
1682:December 2019
1676:
1672:
1669:This section
1667:
1664:
1660:
1659:
1653:
1650:
1642:
1640:
1638:
1637:tight closure
1633:
1619:
1599:
1573:
1569:
1562:
1559:
1554:
1550:
1540:
1535:
1531:
1525:
1522:
1519:
1515:
1511:
1505:
1502:
1499:
1496:
1493:
1490:
1485:
1481:
1477:
1474:
1471:
1468:
1465:
1459:
1447:
1443:
1440:
1430:
1429:
1428:
1414:
1394:
1391:
1388:
1381:Given ideals
1379:
1365:
1345:
1323:
1319:
1296:
1292:
1269:
1265:
1261:
1258:
1255:
1247:
1243:
1232:
1223:
1219:
1216:This section
1214:
1211:
1207:
1206:
1200:
1198:
1196:
1192:
1187:
1185:
1181:
1177:
1173:
1168:
1166:
1165:
1158:
1156:
1152:
1148:
1132:
1129:
1124:
1121:
1117:
1112:
1092:
1084:
1081:(that is, an
1066:
1063:
1059:
1051:
1035:
1015:
1007:
1004:
996:
994:
992:
990:
970:
933:
893:
883:
880:
877:
869:
863:
860:
857:
851:
843:
805:
798:. Hence, the
785:
765:
742:
739:
734:
724:
721:
718:
710:
702:
699:
693:
687:
684:
681:
675:
644:
641:
638:
632:
628:
624:
621:
615:
612:
609:
604:
600:
596:
593:
586:
585:
584:
570:
563:
548:
528:
523:
519:
515:
512:
490:
487:
483:
479:
474:
464:
439:
431:
415:
412:
409:
406:
381:
378:
369:
355:
335:
332:
329:
326:
318:
314:
298:
295:
292:
269:
266:
263:
258:
254:
250:
247:
240:
239:
238:
236:
232:
228:
224:
220:
216:
212:
206:
198:
196:
182:
174:
170:
169:Jacobson ring
154:
134:
126:
110:
99:
88:
79:
75:
72:This section
70:
67:
63:
62:
56:
54:
52:
47:
45:
41:
37:
33:
29:
22:
2166:, retrieved
2162:the original
2134:
2113:
2093:
2020:
1982:
1863:
1859:
1695:
1692:
1679:
1675:adding to it
1670:
1634:
1591:
1427:, the ideal
1380:
1239:
1226:
1222:adding to it
1217:
1188:
1183:
1176:Picard group
1171:
1169:
1162:
1159:
1154:
1146:
1105:) such that
1082:
1005:
1000:
988:
842:metric space
778:with center
667:
429:
370:
316:
312:
284:
234:
230:
226:
222:
218:
214:
210:
208:
101:
82:
78:adding to it
73:
48:
32:ideal theory
31:
25:
1028:, an ideal
28:mathematics
2179:Categories
2168:2019-11-15
2079:References
1407:in a ring
96:See also:
1992:Γ
1945:
1939:
1934:→
1917:~
1899:Γ
1878:Γ
1834:−
1822:
1790:~
1635:See also
1541:
1516:⋃
1503:≫
1491:⊂
1475:∣
1469:∈
1452:∞
1262:∩
1122:−
1064:−
932:completed
881:−
722:−
711:∣
703:∈
642:−
633:−
488:−
452:, define
264:⊂
2132:(2006),
2092:(1969),
2067:See also
1229:May 2022
1180:spectrum
987:ring of
562:prime to
85:May 2022
2158:2266432
1178:of the
930:can be
2156:
2146:
2120:
2100:
1983:Here,
668:where
36:ideals
1001:In a
505:when
167:is a
2144:ISBN
2118:ISBN
2098:ISBN
1819:Spec
1736:and
1696:Let
1523:>
740:<
296:>
2023:of
1942:Hom
1931:lim
1805:on
1677:.
1532:Ann
1224:.
1182:of
1153:of
233:in
225:of
209:If
127:in
80:.
38:in
26:In
2181::
2154:MR
2152:,
2142:,
2112:,
2088:;
2063:.
1896::=
1639:.
1197:.
1157:.
993:.
541:,
368:.
237:,
30:,
2124:.
2051:I
2031:M
2007:)
2004:M
2001:(
1996:I
1979:.
1967:)
1964:M
1961:,
1956:n
1952:I
1948:(
1926:=
1923:)
1914:M
1908:,
1905:Y
1902:(
1893:)
1890:M
1887:(
1882:I
1864:M
1860:Y
1846:)
1843:I
1840:(
1837:V
1831:)
1828:R
1825:(
1816:=
1813:Y
1787:M
1764:M
1744:I
1724:R
1704:M
1684:)
1680:(
1620:J
1600:I
1577:)
1574:I
1570:/
1566:)
1563:I
1560:+
1555:n
1551:J
1547:(
1544:(
1536:A
1526:0
1520:n
1512:=
1509:}
1506:0
1500:n
1497:,
1494:I
1486:n
1482:J
1478:f
1472:A
1466:f
1463:{
1460:=
1457:)
1448:J
1444::
1441:I
1438:(
1415:A
1395:J
1392:,
1389:I
1366:I
1346:I
1324:j
1320:Q
1297:i
1293:Q
1270:i
1266:Q
1259:=
1256:I
1231:)
1227:(
1184:A
1172:A
1155:A
1147:K
1133:A
1130:=
1125:1
1118:I
1113:I
1093:K
1083:A
1067:1
1060:I
1036:I
1016:K
1006:A
989:p
971:p
966:Z
943:Z
917:Z
894:p
889:|
884:y
878:x
874:|
870:=
867:)
864:y
861:,
858:x
855:(
852:d
827:Z
806:p
786:x
766:r
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743:r
735:p
730:|
725:x
719:z
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700:z
697:{
694:=
691:)
688:r
685:,
682:x
679:(
676:B
653:)
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645:1
639:n
636:(
629:p
625:,
622:x
619:(
616:B
613:=
610:A
605:n
601:p
597:+
594:x
571:p
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529:y
524:n
520:p
516:=
513:x
491:n
484:p
480:=
475:p
470:|
465:x
461:|
440:x
430:p
416:A
413:p
410:=
407:I
386:Z
382:=
379:A
356:a
336:A
333:a
330:=
327:I
317:a
313:I
299:0
293:n
270:.
267:U
259:n
255:I
251:+
248:x
235:U
231:x
227:A
223:U
219:A
215:A
211:I
183:A
155:A
135:A
111:A
87:)
83:(
23:.
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