971:
747:
1104:
114:
Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.
1163:
562:
828:
1876:
784:
663:
1539:
603:
277:
1701:
303:
1636:
179:
239:
209:
1574:
1004:
881:
668:
1796:
1745:
1598:
1486:
1462:
1438:
1414:
1390:
1028:
876:
852:
627:
331:
146:
506:
1343:
1822:
1772:
1721:
1033:
2080:
1109:
511:
789:
1827:
1962:
752:
632:
2091:(in French), MĂ©mor. Sci. Math., Fasc. CLIV. Gauthier-Villars & Cie, Editeur -Imprimeur-Libraire, Paris, p. 119,
1491:
567:
244:
1230:
1641:
2132:
282:
2127:
1603:
151:
966:{\displaystyle ({\overline {x}})\cap ({\overline {x}}^{2},{\overline {x}}{\overline {z}},{\overline {y}})}
214:
184:
103:, that is, can be written as an intersection of finitely many primary ideals. This result is known as the
1548:
742:{\displaystyle {\overline {x}}{\overline {y}}={\overline {z}}^{2}\in {\mathfrak {p}}^{2}={\mathfrak {q}}}
1542:
976:
104:
1777:
1726:
1579:
1467:
1443:
1419:
1395:
1371:
1009:
857:
833:
608:
312:
127:
1600:-primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal
434:
1315:
1979:
Chatters, A. W.; Hajarnavis, C. R. (1971), "Non-commutative rings with primary decomposition",
1350:
431:
On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if
410:
95:
The notion of primary ideals is important in commutative ring theory because every ideal of a
2109:
100:
1801:
2096:
2074:
2046:
2008:
1750:
1903:
See the references to
Chatters–Hajarnavis, Goldman, Gorton–Heatherly, and Lesieur–Croisot.
8:
398:
380:
306:
32:
28:
1706:
2062:
2034:
2029:
1996:
1958:
108:
2053:
Gorton, Christine; Heatherly, Henry (2006), "Generalized primary rings and ideals",
1099:{\displaystyle ({\overline {x}}^{2},{\overline {x}}{\overline {z}},{\overline {y}})}
2024:
1988:
1950:
1703:
yields the zero ideal, which in this case is not primary (because the zero divisor
74:
39:
2092:
2070:
2042:
2004:
1305:
376:
96:
1946:
2121:
2066:
2038:
2000:
1992:
375:
is primary, and moreover an ideal is prime if and only if it is primary and
1464:-primary; since for example, in a Noetherian local ring with maximal ideal
387:
345:
124:
The definition can be rephrased in a more symmetric manner: a proper ideal
89:
372:
24:
17:
1723:
is not nilpotent). In fact, in a
Noetherian ring, a nonempty product of
2052:
1265:, but at least they contain a power of P; for example the ideal (
1158:{\displaystyle ({\overline {x}},{\overline {y}},{\overline {z}})}
557:{\displaystyle {\mathfrak {p}}=({\overline {x}},{\overline {z}})}
1978:
356:
is nilpotent. (Compare this to the case of prime ideals, where
1894:
To be precise, one usually uses this fact to prove the theorem.
1249:
is a maximal prime ideal, then any ideal containing a power of
823:{\displaystyle {\overline {y}}^{n}\not \in {\mathfrak {q}}}
1912:
For the proof of the second part see the article of Fuchs.
2015:
Goldman, Oscar (1969), "Rings and modules of quotients",
1871:{\displaystyle {\mathfrak {p}}^{n}\subset \cap _{i}Q_{i}}
779:{\displaystyle {\overline {x}}\not \in {\mathfrak {q}}}
658:{\displaystyle {\sqrt {\mathfrak {q}}}={\mathfrak {p}}}
2086:
1830:
1804:
1780:
1753:
1729:
1709:
1644:
1606:
1582:
1551:
1494:
1470:
1446:
1422:
1398:
1374:
1318:
1112:
1036:
1012:
979:
884:
860:
836:
792:
755:
671:
635:
611:
570:
514:
437:
315:
285:
247:
217:
187:
154:
130:
1534:{\displaystyle \cap _{n>0}{\mathfrak {m}}^{n}=0}
598:{\displaystyle {\mathfrak {q}}={\mathfrak {p}}^{2}}
1870:
1816:
1798:-primary if and only if there exists some integer
1790:
1766:
1739:
1715:
1695:
1630:
1592:
1568:
1533:
1480:
1456:
1432:
1408:
1384:
1337:
1157:
1098:
1022:
998:
965:
870:
846:
822:
778:
741:
657:
621:
597:
556:
500:
325:
297:
271:
233:
203:
173:
140:
2119:
1945:
2014:
854:is not primary. The primary decomposition of
360:is prime if and only if every zero divisor in
272:{\displaystyle x,y\in {\sqrt {\mathfrak {q}}}}
1690:
1668:
1625:
1613:
1696:{\displaystyle K/\langle x^{2},xy\rangle }
118:
2028:
298:{\displaystyle {\sqrt {\mathfrak {q}}}}
73: > 0. For example, in the
2120:
1631:{\displaystyle m=\langle x,y\rangle }
174:{\displaystyle xy\in {\mathfrak {q}}}
2089:Algèbre noethérienne non commutative
1981:The Quarterly Journal of Mathematics
1970:
1930:
1416:-primary but an infinite product of
234:{\displaystyle y\in {\mathfrak {q}}}
204:{\displaystyle x\in {\mathfrak {q}}}
1955:Introduction to Commutative Algebra
1834:
1783:
1732:
1585:
1569:{\displaystyle {\mathfrak {m}}^{n}}
1555:
1514:
1473:
1449:
1425:
1401:
1377:
1015:
863:
839:
815:
771:
734:
718:
650:
639:
614:
584:
573:
517:
318:
289:
263:
226:
196:
166:
133:
13:
1312:a prime ideal, then the kernel of
1261:-primary ideals need be powers of
14:
2144:
2103:
2087:Lesieur, L.; Croisot, R. (1963),
1921:Atiyah–Macdonald, Corollary 10.21
999:{\displaystyle ({\overline {x}})}
111:of a Noetherian ring is primary.
344:is primary if and only if every
2113:at Encyclopaedia of Mathematics
1791:{\displaystyle {\mathfrak {p}}}
1740:{\displaystyle {\mathfrak {p}}}
1593:{\displaystyle {\mathfrak {m}}}
1481:{\displaystyle {\mathfrak {m}}}
1457:{\displaystyle {\mathfrak {p}}}
1433:{\displaystyle {\mathfrak {p}}}
1409:{\displaystyle {\mathfrak {p}}}
1385:{\displaystyle {\mathfrak {p}}}
1023:{\displaystyle {\mathfrak {p}}}
871:{\displaystyle {\mathfrak {q}}}
847:{\displaystyle {\mathfrak {q}}}
622:{\displaystyle {\mathfrak {p}}}
409:, and this ideal is called the
326:{\displaystyle {\mathfrak {q}}}
141:{\displaystyle {\mathfrak {q}}}
1957:, Westview Press, p. 50,
1924:
1915:
1906:
1897:
1888:
1660:
1648:
1322:
1152:
1113:
1093:
1037:
993:
980:
960:
904:
898:
885:
551:
525:
501:{\displaystyle R=k/(xy-z^{2})}
495:
473:
465:
447:
1:
1939:
1368:A finite nonempty product of
1361:, is the intersection of all
1190:-primary ideal: all elements
405:is necessarily a prime ideal
397:is a primary ideal, then the
2030:10.1016/0021-8693(69)90004-0
1882:
1147:
1134:
1121:
1088:
1075:
1065:
1046:
988:
955:
942:
932:
913:
893:
799:
761:
701:
687:
677:
546:
533:
7:
1440:-primary ideals may not be
10:
2149:
1933:, Ch. IV, § 2, Exercise 3.
1543:Krull intersection theorem
1338:{\displaystyle A\to A_{P}}
1218:-primary ideal containing
1168:An ideal whose radical is
15:
1297:, however it contains P².
383:in the commutative case).
1293:, but is not a power of
1186:contained in a smallest
148:is primary if, whenever
84:) is a primary ideal if
16:Not to be confused with
1947:Atiyah, Michael Francis
1277:-primary for the ideal
386:Every primary ideal is
119:Examples and properties
1872:
1818:
1817:{\displaystyle n>0}
1792:
1768:
1741:
1717:
1697:
1632:
1594:
1570:
1535:
1482:
1458:
1434:
1410:
1386:
1339:
1172:, however, is primary.
1159:
1100:
1024:
1000:
967:
872:
848:
824:
780:
743:
659:
623:
599:
558:
502:
411:associated prime ideal
327:
299:
273:
235:
205:
175:
142:
105:Lasker–Noether theorem
65:is also an element of
2055:Mathematica Pannonica
1993:10.1093/qmath/22.1.73
1873:
1819:
1793:
1769:
1767:{\displaystyle Q_{i}}
1742:
1718:
1698:
1633:
1595:
1571:
1536:
1483:
1459:
1435:
1411:
1387:
1340:
1160:
1101:
1025:
1001:
968:
873:
849:
830:for all n > 0, so
825:
781:
744:
660:
624:
600:
559:
503:
417:. In this situation,
328:
300:
274:
236:
206:
176:
143:
101:primary decomposition
2133:Ideals (ring theory)
1828:
1802:
1778:
1751:
1727:
1707:
1642:
1604:
1580:
1549:
1492:
1468:
1444:
1420:
1396:
1372:
1316:
1110:
1034:
1010:
977:
882:
858:
834:
790:
753:
669:
633:
609:
568:
512:
435:
313:
283:
245:
215:
185:
152:
128:
2128:Commutative algebra
1973:Algèbre commutative
1392:-primary ideals is
1209: ∉
1199: ∈
107:. Consequently, an
29:commutative algebra
2017:Journal of Algebra
1868:
1814:
1788:
1764:
1737:
1713:
1693:
1638:of the local ring
1628:
1590:
1566:
1531:
1478:
1454:
1430:
1406:
1382:
1335:
1257:-primary. Not all
1155:
1096:
1020:
996:
963:
868:
844:
820:
776:
739:
655:
619:
595:
554:
498:
368:is actually zero.)
323:
295:
269:
231:
201:
171:
138:
2083:, Ladislas Fuchs
1983:, Second Series,
1964:978-0-201-40751-8
1716:{\displaystyle y}
1150:
1137:
1124:
1091:
1078:
1068:
1049:
991:
958:
945:
935:
916:
896:
802:
764:
704:
690:
680:
643:
549:
536:
293:
267:
109:irreducible ideal
53:is an element of
2140:
2099:
2081:On primal ideals
2077:
2049:
2032:
2011:
1975:
1967:
1934:
1928:
1922:
1919:
1913:
1910:
1904:
1901:
1895:
1892:
1877:
1875:
1874:
1869:
1867:
1866:
1857:
1856:
1844:
1843:
1838:
1837:
1823:
1821:
1820:
1815:
1797:
1795:
1794:
1789:
1787:
1786:
1773:
1771:
1770:
1765:
1763:
1762:
1747:-primary ideals
1746:
1744:
1743:
1738:
1736:
1735:
1722:
1720:
1719:
1714:
1702:
1700:
1699:
1694:
1680:
1679:
1667:
1637:
1635:
1634:
1629:
1599:
1597:
1596:
1591:
1589:
1588:
1575:
1573:
1572:
1567:
1565:
1564:
1559:
1558:
1540:
1538:
1537:
1532:
1524:
1523:
1518:
1517:
1510:
1509:
1487:
1485:
1484:
1479:
1477:
1476:
1463:
1461:
1460:
1455:
1453:
1452:
1439:
1437:
1436:
1431:
1429:
1428:
1415:
1413:
1412:
1407:
1405:
1404:
1391:
1389:
1388:
1383:
1381:
1380:
1365:-primary ideals.
1344:
1342:
1341:
1336:
1334:
1333:
1237:
1227:
1223:
1217:
1213:
1203:
1193:
1189:
1182:
1178:
1164:
1162:
1161:
1156:
1151:
1143:
1138:
1130:
1125:
1117:
1105:
1103:
1102:
1097:
1092:
1084:
1079:
1071:
1069:
1061:
1056:
1055:
1050:
1042:
1029:
1027:
1026:
1021:
1019:
1018:
1005:
1003:
1002:
997:
992:
984:
972:
970:
969:
964:
959:
951:
946:
938:
936:
928:
923:
922:
917:
909:
897:
889:
877:
875:
874:
869:
867:
866:
853:
851:
850:
845:
843:
842:
829:
827:
826:
821:
819:
818:
809:
808:
803:
795:
785:
783:
782:
777:
775:
774:
765:
757:
748:
746:
745:
740:
738:
737:
728:
727:
722:
721:
711:
710:
705:
697:
691:
683:
681:
673:
664:
662:
661:
656:
654:
653:
644:
642:
637:
628:
626:
625:
620:
618:
617:
604:
602:
601:
596:
594:
593:
588:
587:
577:
576:
563:
561:
560:
555:
550:
542:
537:
529:
521:
520:
507:
505:
504:
499:
494:
493:
472:
332:
330:
329:
324:
322:
321:
304:
302:
301:
296:
294:
292:
287:
278:
276:
275:
270:
268:
266:
261:
240:
238:
237:
232:
230:
229:
210:
208:
207:
202:
200:
199:
180:
178:
177:
172:
170:
169:
147:
145:
144:
139:
137:
136:
75:ring of integers
40:commutative ring
2148:
2147:
2143:
2142:
2141:
2139:
2138:
2137:
2118:
2117:
2106:
1965:
1951:Macdonald, I.G.
1942:
1937:
1929:
1925:
1920:
1916:
1911:
1907:
1902:
1898:
1893:
1889:
1885:
1862:
1858:
1852:
1848:
1839:
1833:
1832:
1831:
1829:
1826:
1825:
1803:
1800:
1799:
1782:
1781:
1779:
1776:
1775:
1758:
1754:
1752:
1749:
1748:
1731:
1730:
1728:
1725:
1724:
1708:
1705:
1704:
1675:
1671:
1663:
1643:
1640:
1639:
1605:
1602:
1601:
1584:
1583:
1581:
1578:
1577:
1560:
1554:
1553:
1552:
1550:
1547:
1546:
1519:
1513:
1512:
1511:
1499:
1495:
1493:
1490:
1489:
1472:
1471:
1469:
1466:
1465:
1448:
1447:
1445:
1442:
1441:
1424:
1423:
1421:
1418:
1417:
1400:
1399:
1397:
1394:
1393:
1376:
1375:
1373:
1370:
1369:
1345:, the map from
1329:
1325:
1317:
1314:
1313:
1306:Noetherian ring
1289:) in the ring
1235:
1225:
1219:
1215:
1214:. The smallest
1205:
1195:
1191:
1187:
1180:
1176:
1142:
1129:
1116:
1111:
1108:
1107:
1083:
1070:
1060:
1051:
1041:
1040:
1035:
1032:
1031:
1014:
1013:
1011:
1008:
1007:
983:
978:
975:
974:
950:
937:
927:
918:
908:
907:
888:
883:
880:
879:
862:
861:
859:
856:
855:
838:
837:
835:
832:
831:
814:
813:
804:
794:
793:
791:
788:
787:
770:
769:
756:
754:
751:
750:
733:
732:
723:
717:
716:
715:
706:
696:
695:
682:
672:
670:
667:
666:
649:
648:
638:
636:
634:
631:
630:
613:
612:
610:
607:
606:
589:
583:
582:
581:
572:
571:
569:
566:
565:
541:
528:
516:
515:
513:
510:
509:
489:
485:
468:
436:
433:
432:
336:A proper ideal
317:
316:
314:
311:
310:
288:
286:
284:
281:
280:
262:
260:
246:
243:
242:
225:
224:
216:
213:
212:
195:
194:
186:
183:
182:
165:
164:
153:
150:
149:
132:
131:
129:
126:
125:
121:
97:Noetherian ring
27:, specifically
21:
12:
11:
5:
2146:
2136:
2135:
2130:
2116:
2115:
2105:
2104:External links
2102:
2101:
2100:
2084:
2078:
2050:
2012:
1976:
1968:
1963:
1941:
1938:
1936:
1935:
1923:
1914:
1905:
1896:
1886:
1884:
1881:
1880:
1879:
1865:
1861:
1855:
1851:
1847:
1842:
1836:
1813:
1810:
1807:
1785:
1761:
1757:
1734:
1712:
1692:
1689:
1686:
1683:
1678:
1674:
1670:
1666:
1662:
1659:
1656:
1653:
1650:
1647:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1587:
1563:
1557:
1530:
1527:
1522:
1516:
1508:
1505:
1502:
1498:
1475:
1451:
1427:
1403:
1379:
1366:
1332:
1328:
1324:
1321:
1298:
1281: = (
1243:
1242:
1241:
1240:
1239:
1231:symbolic power
1224:is called the
1173:
1154:
1149:
1146:
1141:
1136:
1133:
1128:
1123:
1120:
1115:
1095:
1090:
1087:
1082:
1077:
1074:
1067:
1064:
1059:
1054:
1048:
1045:
1039:
1017:
995:
990:
987:
982:
962:
957:
954:
949:
944:
941:
934:
931:
926:
921:
915:
912:
906:
903:
900:
895:
892:
887:
865:
841:
817:
812:
807:
801:
798:
773:
768:
763:
760:
736:
731:
726:
720:
714:
709:
703:
700:
694:
689:
686:
679:
676:
665:, but we have
652:
647:
641:
616:
592:
586:
580:
575:
553:
548:
545:
540:
535:
532:
527:
524:
519:
497:
492:
488:
484:
481:
478:
475:
471:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
421:is said to be
391:
384:
369:
334:
320:
291:
265:
259:
256:
253:
250:
228:
223:
220:
198:
193:
190:
168:
163:
160:
157:
135:
120:
117:
45:is said to be
9:
6:
4:
3:
2:
2145:
2134:
2131:
2129:
2126:
2125:
2123:
2114:
2112:
2111:Primary ideal
2108:
2107:
2098:
2094:
2090:
2085:
2082:
2079:
2076:
2072:
2068:
2064:
2060:
2056:
2051:
2048:
2044:
2040:
2036:
2031:
2026:
2022:
2018:
2013:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1977:
1974:
1969:
1966:
1960:
1956:
1952:
1948:
1944:
1943:
1932:
1927:
1918:
1909:
1900:
1891:
1887:
1863:
1859:
1853:
1849:
1845:
1840:
1811:
1808:
1805:
1759:
1755:
1710:
1687:
1684:
1681:
1676:
1672:
1664:
1657:
1654:
1651:
1645:
1622:
1619:
1616:
1610:
1607:
1561:
1545:) where each
1544:
1528:
1525:
1520:
1506:
1503:
1500:
1496:
1367:
1364:
1360:
1356:
1352:
1348:
1330:
1326:
1319:
1311:
1307:
1303:
1299:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1233:
1232:
1222:
1212:
1208:
1202:
1198:
1185:
1179:with radical
1175:Every ideal
1174:
1171:
1167:
1166:
1144:
1139:
1131:
1126:
1118:
1085:
1080:
1072:
1062:
1057:
1052:
1043:
1030:-primary and
985:
952:
947:
939:
929:
924:
919:
910:
901:
890:
810:
805:
796:
766:
758:
729:
724:
712:
707:
698:
692:
684:
674:
645:
629:is prime and
590:
578:
543:
538:
530:
522:
490:
486:
482:
479:
476:
469:
462:
459:
456:
453:
450:
444:
441:
438:
430:
429:
427:
425:
420:
416:
412:
408:
404:
400:
396:
392:
389:
385:
382:
381:radical ideal
379:(also called
378:
374:
370:
367:
363:
359:
355:
351:
347:
343:
339:
335:
308:
257:
254:
251:
248:
221:
218:
191:
188:
161:
158:
155:
123:
122:
116:
112:
110:
106:
102:
98:
93:
91:
87:
83:
79:
76:
72:
68:
64:
60:
56:
52:
48:
44:
41:
37:
34:
30:
26:
19:
2110:
2088:
2061:(1): 17–28,
2058:
2054:
2020:
2016:
1984:
1980:
1972:
1954:
1926:
1917:
1908:
1899:
1890:
1362:
1358:
1354:
1351:localization
1346:
1309:
1301:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1229:
1220:
1210:
1206:
1200:
1196:
1183:
1169:
423:
422:
418:
414:
406:
402:
394:
365:
361:
357:
353:
349:
346:zero divisor
341:
337:
305:denotes the
113:
94:
90:prime number
85:
81:
77:
70:
66:
62:
58:
54:
50:
49:if whenever
46:
42:
35:
22:
373:prime ideal
69:, for some
31:, a proper
25:mathematics
18:Prime ideal
2122:Categories
1971:Bourbaki,
1940:References
1824:such that
1194:such that
1165:-primary.
181:, we have
2067:0865-2090
2039:0021-8693
2023:: 10–47,
2001:0033-5606
1987:: 73–83,
1883:Footnotes
1850:∩
1846:⊂
1691:⟩
1669:⟨
1626:⟩
1614:⟨
1497:∩
1323:→
1204:for some
1148:¯
1135:¯
1122:¯
1089:¯
1076:¯
1066:¯
1047:¯
989:¯
956:¯
943:¯
933:¯
914:¯
902:∩
894:¯
800:¯
762:¯
713:∈
702:¯
688:¯
678:¯
547:¯
534:¯
483:−
377:semiprime
279:. (Here
258:∈
222:∈
192:∈
162:∈
1953:(1969),
1931:Bourbaki
811:∉
767:∉
426:-primary
2097:0155861
2075:2215638
2047:0245608
2009:0286822
1349:to the
1285:,
1269:,
1170:maximal
973:; here
605:, then
399:radical
307:radical
47:primary
2095:
2073:
2065:
2045:
2037:
2007:
1999:
1961:
786:, and
564:, and
388:primal
99:has a
1304:is a
1273:) is
88:is a
57:then
38:of a
33:ideal
2063:ISSN
2035:ISSN
1997:ISSN
1959:ISBN
1809:>
1504:>
1308:and
371:Any
2025:doi
1989:doi
1774:is
1576:is
1357:at
1353:of
1300:If
1253:is
1245:If
1234:of
1228:th
1106:is
1006:is
878:is
413:of
401:of
393:If
348:in
340:of
309:of
241:or
211:or
80:, (
61:or
23:In
2124::
2093:MR
2071:MR
2069:,
2059:17
2057:,
2043:MR
2041:,
2033:,
2021:13
2019:,
2005:MR
2003:,
1995:,
1985:22
1949:;
1488:,
1197:ax
1184:is
749:,
508:,
428:.
333:.)
92:.
51:xy
2027::
1991::
1878:.
1864:i
1860:Q
1854:i
1841:n
1835:p
1812:0
1806:n
1784:p
1760:i
1756:Q
1733:p
1711:y
1688:y
1685:x
1682:,
1677:2
1673:x
1665:/
1661:]
1658:y
1655:,
1652:x
1649:[
1646:K
1623:y
1620:,
1617:x
1611:=
1608:m
1586:m
1562:n
1556:m
1541:(
1529:0
1526:=
1521:n
1515:m
1507:0
1501:n
1474:m
1450:p
1426:p
1402:p
1378:p
1363:P
1359:P
1355:A
1347:A
1331:P
1327:A
1320:A
1310:P
1302:A
1295:P
1291:k
1287:y
1283:x
1279:P
1275:P
1271:y
1267:x
1263:P
1259:P
1255:P
1251:P
1247:P
1238:.
1236:P
1226:n
1221:P
1216:P
1211:P
1207:x
1201:Q
1192:a
1188:P
1181:P
1177:Q
1153:)
1145:z
1140:,
1132:y
1127:,
1119:x
1114:(
1094:)
1086:y
1081:,
1073:z
1063:x
1058:,
1053:2
1044:x
1038:(
1016:p
994:)
986:x
981:(
961:)
953:y
948:,
940:z
930:x
925:,
920:2
911:x
905:(
899:)
891:x
886:(
864:q
840:q
816:q
806:n
797:y
772:q
759:x
735:q
730:=
725:2
719:p
708:2
699:z
693:=
685:y
675:x
651:p
646:=
640:q
615:p
591:2
585:p
579:=
574:q
552:)
544:z
539:,
531:x
526:(
523:=
518:p
496:)
491:2
487:z
480:y
477:x
474:(
470:/
466:]
463:z
460:,
457:y
454:,
451:x
448:[
445:k
442:=
439:R
424:P
419:Q
415:Q
407:P
403:Q
395:Q
390:.
366:P
364:/
362:R
358:P
354:Q
352:/
350:R
342:R
338:Q
319:q
290:q
264:q
255:y
252:,
249:x
227:q
219:y
197:q
189:x
167:q
159:y
156:x
134:q
86:p
82:p
78:Z
71:n
67:Q
63:y
59:x
55:Q
43:A
36:Q
20:.
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