211:
1162:
1090:
1240:
882:
738:
969:
598:
669:
369:
1117:
324:
257:
53:
105:
515:
469:
426:
297:
1311:
1266:
802:
1030:
1001:
1164:; i.e., the integral closure of an ideal is integrally closed. It also follows that the integral closure of a homogeneous ideal is homogeneous.
1398:
1373:
113:
1122:
1035:
1384:
1462:
1186:
1452:
971:
can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of
379:
1326:
823:
695:
1390:
903:
375:
520:
1280:
1314:
677:(e.g., prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed.
603:
740:. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
341:
1457:
1095:
302:
235:
31:
70:
474:
431:
394:
270:
1290:
1394:
1369:
1245:
217:
1408:
1404:
746:
1361:
1006:
974:
817:
1446:
1432:
1380:
674:
1412:
898:
1389:, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK:
1436:
681:
330:(in fact, the integral closure of an ideal is always an ideal; see below.)
1313:
have the same integral closure if and only if they have the same
1368:, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995,
206:{\displaystyle r^{n}+a_{1}r^{n-1}+\cdots +a_{n-1}r+a_{n}=0.}
1157:{\displaystyle {\overline {I}}={\overline {\overline {I}}}}
1366:
Commutative
Algebra with a View Toward Algebraic Geometry
1085:{\displaystyle \oplus _{n\geq 0}{\overline {I^{n}}}t^{n}}
374:
The integral closure of an ideal appears in a theorem of
884:. The integral closure of a monomial ideal is monomial.
1293:
1283:(i.e., the completion is equidimensional.). Then two
1248:
1189:
1125:
1098:
1038:
1009:
977:
906:
826:
749:
698:
606:
523:
477:
434:
397:
344:
305:
273:
238:
116:
73:
34:
1235:{\displaystyle {\overline {I^{n+l}}}\subset I^{n+1}}
1305:
1260:
1234:
1156:
1111:
1084:
1024:
995:
963:
876:
796:
732:
663:
592:
509:
463:
420:
363:
318:
291:
251:
205:
99:
47:
877:{\displaystyle X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}}
733:{\displaystyle {\overline {xI}}=x{\overline {I}}}
1386:Integral closure of ideals, rings, and modules
1345:
964:{\displaystyle R=\oplus _{n\geq 0}I^{n}t^{n}}
259:if and only if there is a finitely generated
8:
1167:The following type of results is called the
1279:) be a noetherian local ring. Assume it is
593:{\displaystyle r^{d}+(-x^{di}y^{d(d-i)})=0}
1292:
1247:
1220:
1196:
1190:
1188:
1139:
1126:
1124:
1099:
1097:
1076:
1061:
1055:
1043:
1037:
1008:
976:
955:
945:
929:
905:
866:
861:
856:
841:
836:
831:
825:
785:
766:
748:
720:
699:
697:
640:
627:
611:
605:
560:
547:
528:
522:
498:
485:
476:
449:
439:
433:
399:
398:
396:
351:
343:
306:
304:
272:
239:
237:
191:
169:
144:
134:
121:
115:
91:
78:
72:
35:
33:
1338:
820:if it is generated by monomials; i.e.,
664:{\displaystyle a_{d}=-x^{di}y^{d(d-i)}}
267:, annihilated only by zero, such that
7:
1411:, Reference-idHS2006, archived from
804:be a polynomial ring over a field
14:
364:{\displaystyle I={\overline {I}}}
1271:A theorem of Rees states: let (
1112:{\displaystyle {\overline {I}}}
319:{\displaystyle {\overline {I}}}
252:{\displaystyle {\overline {I}}}
48:{\displaystyle {\overline {I}}}
1019:
1013:
990:
981:
919:
910:
791:
759:
656:
644:
581:
576:
564:
537:
504:
478:
415:
403:
220:of a subring. For example, if
100:{\displaystyle a_{i}\in I^{i}}
1:
510:{\displaystyle (x^{d},y^{d})}
55:, is the set of all elements
1208:
1149:
1145:
1131:
1104:
1067:
725:
709:
517:. It satisfies the equation
464:{\displaystyle x^{i}y^{d-i}}
421:{\displaystyle \mathbb {C} }
380:analytically unramified ring
356:
311:
292:{\displaystyle rM\subset IM}
244:
40:
1479:
1391:Cambridge University Press
1306:{\displaystyle I\subset J}
684:, for any non-zerodivisor
1346:Swanson & Huneke 2006
1383:; Huneke, Craig (2006),
1281:formally equidimensional
224:is a domain, an element
1327:Dedekind–Kummer theorem
1261:{\displaystyle n\geq 0}
63:that are integral over
1307:
1262:
1236:
1179:an ideal generated by
1175:be a regular ring and
1169:Briancon–Skoda theorem
1158:
1113:
1086:
1032:, which is graded, is
1026:
997:
965:
878:
798:
734:
665:
594:
511:
465:
422:
378:that characterizes an
365:
320:
293:
253:
207:
101:
49:
24:of a commutative ring
1308:
1263:
1237:
1159:
1114:
1087:
1027:
998:
966:
879:
799:
735:
666:
595:
512:
466:
423:
366:
321:
294:
254:
216:It is similar to the
208:
102:
50:
1463:Algebraic structures
1291:
1246:
1187:
1123:
1096:
1036:
1007:
975:
904:
824:
747:
696:
604:
521:
475:
432:
395:
342:
303:
271:
236:
114:
71:
32:
1453:Commutative algebra
873:
848:
797:{\displaystyle R=k}
1303:
1258:
1232:
1154:
1109:
1082:
1022:
993:
961:
874:
852:
827:
794:
730:
661:
590:
507:
461:
418:
361:
316:
299:. It follows that
289:
249:
203:
97:
45:
1400:978-0-521-68860-4
1211:
1152:
1148:
1134:
1107:
1092:. In particular,
1070:
1025:{\displaystyle R}
996:{\displaystyle R}
889:Structure results
728:
712:
471:is integral over
359:
336:integrally closed
314:
247:
43:
1470:
1422:
1421:
1420:
1349:
1348:, Theorem 11.3.1
1343:
1312:
1310:
1309:
1304:
1287:-primary ideals
1267:
1265:
1264:
1259:
1241:
1239:
1238:
1233:
1231:
1230:
1212:
1207:
1206:
1191:
1182:
1178:
1163:
1161:
1160:
1155:
1153:
1141:
1140:
1135:
1127:
1119:is an ideal and
1118:
1116:
1115:
1110:
1108:
1100:
1091:
1089:
1088:
1083:
1081:
1080:
1071:
1066:
1065:
1056:
1054:
1053:
1031:
1029:
1028:
1023:
1002:
1000:
999:
994:
970:
968:
967:
962:
960:
959:
950:
949:
940:
939:
883:
881:
880:
875:
872:
871:
870:
860:
847:
846:
845:
835:
803:
801:
800:
795:
790:
789:
771:
770:
739:
737:
736:
731:
729:
721:
713:
708:
700:
671:is in the ideal.
670:
668:
667:
662:
660:
659:
635:
634:
616:
615:
599:
597:
596:
591:
580:
579:
555:
554:
533:
532:
516:
514:
513:
508:
503:
502:
490:
489:
470:
468:
467:
462:
460:
459:
444:
443:
427:
425:
424:
419:
402:
370:
368:
367:
362:
360:
352:
325:
323:
322:
317:
315:
307:
298:
296:
295:
290:
258:
256:
255:
250:
248:
240:
218:integral closure
212:
210:
209:
204:
196:
195:
180:
179:
155:
154:
139:
138:
126:
125:
106:
104:
103:
98:
96:
95:
83:
82:
54:
52:
51:
46:
44:
36:
18:integral closure
16:In algebra, the
1478:
1477:
1473:
1472:
1471:
1469:
1468:
1467:
1443:
1442:
1437:Rees valuations
1429:
1427:Further reading
1418:
1416:
1401:
1379:
1362:Eisenbud, David
1358:
1353:
1352:
1344:
1340:
1335:
1323:
1289:
1288:
1244:
1243:
1216:
1192:
1185:
1184:
1183:elements. Then
1180:
1176:
1121:
1120:
1094:
1093:
1072:
1057:
1039:
1034:
1033:
1005:
1004:
973:
972:
951:
941:
925:
902:
901:
897:be a ring. The
891:
862:
837:
822:
821:
781:
762:
745:
744:
701:
694:
693:
636:
623:
607:
602:
601:
556:
543:
524:
519:
518:
494:
481:
473:
472:
445:
435:
430:
429:
393:
392:
388:
340:
339:
326:is an ideal of
301:
300:
269:
268:
234:
233:
187:
165:
140:
130:
117:
112:
111:
87:
74:
69:
68:
30:
29:
12:
11:
5:
1476:
1474:
1466:
1465:
1460:
1455:
1445:
1444:
1441:
1440:
1428:
1425:
1424:
1423:
1399:
1381:Swanson, Irena
1377:
1357:
1354:
1351:
1350:
1337:
1336:
1334:
1331:
1330:
1329:
1322:
1319:
1302:
1299:
1296:
1257:
1254:
1251:
1229:
1226:
1223:
1219:
1215:
1210:
1205:
1202:
1199:
1195:
1151:
1147:
1144:
1138:
1133:
1130:
1106:
1103:
1079:
1075:
1069:
1064:
1060:
1052:
1049:
1046:
1042:
1021:
1018:
1015:
1012:
992:
989:
986:
983:
980:
958:
954:
948:
944:
938:
935:
932:
928:
924:
921:
918:
915:
912:
909:
890:
887:
886:
885:
869:
865:
859:
855:
851:
844:
840:
834:
830:
793:
788:
784:
780:
777:
774:
769:
765:
761:
758:
755:
752:
741:
727:
724:
719:
716:
711:
707:
704:
688:and any ideal
678:
675:Radical ideals
672:
658:
655:
652:
649:
646:
643:
639:
633:
630:
626:
622:
619:
614:
610:
589:
586:
583:
578:
575:
572:
569:
566:
563:
559:
553:
550:
546:
542:
539:
536:
531:
527:
506:
501:
497:
493:
488:
484:
480:
458:
455:
452:
448:
442:
438:
417:
414:
411:
408:
405:
401:
387:
384:
358:
355:
350:
347:
334:is said to be
313:
310:
288:
285:
282:
279:
276:
246:
243:
214:
213:
202:
199:
194:
190:
186:
183:
178:
175:
172:
168:
164:
161:
158:
153:
150:
147:
143:
137:
133:
129:
124:
120:
94:
90:
86:
81:
77:
67:: there exist
42:
39:
13:
10:
9:
6:
4:
3:
2:
1475:
1464:
1461:
1459:
1456:
1454:
1451:
1450:
1448:
1438:
1434:
1433:Irena Swanson
1431:
1430:
1426:
1415:on 2019-11-15
1414:
1410:
1406:
1402:
1396:
1392:
1388:
1387:
1382:
1378:
1375:
1374:0-387-94268-8
1371:
1367:
1363:
1360:
1359:
1355:
1347:
1342:
1339:
1332:
1328:
1325:
1324:
1320:
1318:
1316:
1300:
1297:
1294:
1286:
1282:
1278:
1274:
1269:
1255:
1252:
1249:
1227:
1224:
1221:
1217:
1213:
1203:
1200:
1197:
1193:
1174:
1170:
1165:
1142:
1136:
1128:
1101:
1077:
1073:
1062:
1058:
1050:
1047:
1044:
1040:
1016:
1010:
987:
984:
978:
956:
952:
946:
942:
936:
933:
930:
926:
922:
916:
913:
907:
900:
896:
888:
867:
863:
857:
853:
849:
842:
838:
832:
828:
819:
815:
811:
807:
786:
782:
778:
775:
772:
767:
763:
756:
753:
750:
742:
722:
717:
714:
705:
702:
691:
687:
683:
679:
676:
673:
653:
650:
647:
641:
637:
631:
628:
624:
620:
617:
612:
608:
587:
584:
573:
570:
567:
561:
557:
551:
548:
544:
540:
534:
529:
525:
499:
495:
491:
486:
482:
456:
453:
450:
446:
440:
436:
412:
409:
406:
390:
389:
385:
383:
381:
377:
372:
353:
348:
345:
337:
333:
329:
308:
286:
283:
280:
277:
274:
266:
262:
241:
231:
227:
223:
219:
200:
197:
192:
188:
184:
181:
176:
173:
170:
166:
162:
159:
156:
151:
148:
145:
141:
135:
131:
127:
122:
118:
110:
109:
108:
92:
88:
84:
79:
75:
66:
62:
58:
37:
28:, denoted by
27:
23:
19:
1417:, retrieved
1413:the original
1385:
1365:
1341:
1315:multiplicity
1284:
1276:
1272:
1270:
1172:
1168:
1166:
899:Rees algebra
894:
892:
813:
809:
805:
689:
685:
373:
335:
331:
327:
264:
260:
229:
225:
221:
215:
64:
60:
56:
25:
21:
20:of an ideal
17:
15:
1458:Ring theory
808:. An ideal
682:normal ring
232:belongs to
1447:Categories
1419:2013-07-12
1356:References
816:is called
107:such that
1298:⊂
1253:≥
1214:⊂
1209:¯
1150:¯
1146:¯
1132:¯
1105:¯
1068:¯
1048:≥
1041:⊕
934:≥
927:⊕
850:⋯
776:…
726:¯
710:¯
651:−
621:−
571:−
541:−
454:−
357:¯
312:¯
281:⊂
245:¯
174:−
160:⋯
149:−
85:∈
41:¯
1321:See also
1242:for any
818:monomial
600:, where
386:Examples
263:-module
1409:2266432
1407:
1397:
1372:
1171:: let
1333:Notes
680:In a
1395:ISBN
1370:ISBN
893:Let
743:Let
376:Rees
1003:in
812:in
391:In
338:if
228:in
59:in
1449::
1435:,
1405:MR
1403:,
1393:,
1364:,
1317:.
1275:,
1268:.
692:,
428:,
382:.
371:.
201:0.
1439:.
1376:.
1301:J
1295:I
1285:m
1277:m
1273:R
1256:0
1250:n
1228:1
1225:+
1222:n
1218:I
1204:l
1201:+
1198:n
1194:I
1181:l
1177:I
1173:R
1143:I
1137:=
1129:I
1102:I
1078:n
1074:t
1063:n
1059:I
1051:0
1045:n
1020:]
1017:t
1014:[
1011:R
991:]
988:t
985:I
982:[
979:R
957:n
953:t
947:n
943:I
937:0
931:n
923:=
920:]
917:t
914:I
911:[
908:R
895:R
868:n
864:a
858:n
854:X
843:1
839:a
833:1
829:X
814:R
810:I
806:k
792:]
787:n
783:X
779:,
773:,
768:1
764:X
760:[
757:k
754:=
751:R
723:I
718:x
715:=
706:I
703:x
690:I
686:x
657:)
654:i
648:d
645:(
642:d
638:y
632:i
629:d
625:x
618:=
613:d
609:a
588:0
585:=
582:)
577:)
574:i
568:d
565:(
562:d
558:y
552:i
549:d
545:x
538:(
535:+
530:d
526:r
505:)
500:d
496:y
492:,
487:d
483:x
479:(
457:i
451:d
447:y
441:i
437:x
416:]
413:y
410:,
407:x
404:[
400:C
354:I
349:=
346:I
332:I
328:R
309:I
287:M
284:I
278:M
275:r
265:M
261:R
242:I
230:R
226:r
222:R
198:=
193:n
189:a
185:+
182:r
177:1
171:n
167:a
163:+
157:+
152:1
146:n
142:r
136:1
132:a
128:+
123:n
119:r
93:i
89:I
80:i
76:a
65:I
61:R
57:r
38:I
26:R
22:I
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