84:
74:
53:
22:
672:
In the 'Properties' section (1.3), before the bullet point about the bijection between prime ideals in the ring and in the localisation, there should be a bullet point about the bijection between ordinary ideals of the ring and the localisation; the bijection of the prime ideals is then a restriction
199:
A recent edit suggested inverting a multiplicative system containing an ideal. However, every ideal contains 0, so the localization at any multiplicative system containing an ideal is the zero ring. I think they just meant the semigroup containing a specific element, which might as well be written
233:
I have a strange feeling the citation at the bottom should be Lang's
Algebraic Number Theory. I know of no book by him entitled "Analytic Number Theory," and furthermore, the information in this article falls under algebraic number theory, not analytic.
698:
I agree it’s a very good idea to mention the bijection doesn’t restrict to maximal ideals (counterexample I can think: R = 2-dimensional ring, finitely generated as an algebra over a field, and p height-one prime, then pR_p is maximal while p isn’t.) —-
179:
I think the author has the wrong definition of "total ring of fractions," which I believe is a very specific localization, namely the localization of a ring with respect to the multiplicatively closed set of all non-zero-divisors in that ring.
987:
Also, Taku's example may be generalized as: If p is any non-maximal prime ideal, then pR_p is maximal. Another example is: if R is a local ring of dimension higher that one, and S contains a non-unit non nilpotent element, then
695:
Are you sure there is such a bijection for ideals not just prime ideals? If I recall, there is such one for primary ideals but not sure about ordinary ideals. In any case, one need a ref for such a statement.
504:
140:
891:
793:
1295:
1490:
848:
1355:
1329:
1233:
746:
1519:
1418:
1158:
1125:
1052:
1019:
968:
924:
1188:
819:
377:
339:
313:
1259:
1539:
1458:
1438:
1593:
677:
restrict to a bijection of maximal ideals. The reason I have not made this edit myself is because I do not know a counterexample for the maximal ideals.
130:
222:
I believe that the end of the article contains an error. Micro local analysis has nothing to do with (micro) localization, as far as I understand.
106:
1588:
241:
270:
577:
Oh wait, upon rereading it I see of course the author did make that mistaken claim about the annihilator. I went ahead and took it out.
508:
97:
58:
181:
165:
of course you're right on the dyadic fractions. Concerning etymology, does the word come from turning rings into local rings? -
385:
263:
First of the properties listed ($ S^{-1}R = 0$ iff $ 0 \in S$ ) appears to be wrong. S may also contain nilpotent element.
33:
499:= 0} in the section "For general commutative rings" is wrong. Any element of Ann(S) annihilates all of S (compare with
1071:
686:
500:
380:
563:
The problem seems to be a disconnect between this statement and the use of Ann(S) in the rest of the paragraph.
245:
274:
616:
162:
It seems that integers are being embedded into dyadic fractions, contrary to what is stated in the article.
673:
of this bijection. There should also be a bullet point afterwards saying that the bijection between ideals
704:
682:
642:
185:
166:
205:
39:
678:
638:
83:
1381:
1372:
Did I miss something ? I think the usual theorem states that this is supposed to be an isomorphism of
266:
237:
21:
1574:
1566:
1554:
1385:
934:. This is an immediate consequence of the following result, which should be added to the article:
661:
646:
620:
586:
572:
534:
512:
468:
453:
445:
390:
278:
249:
209:
189:
169:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1550:
1377:
1067:
853:
755:
657:
612:
582:
568:
530:
464:
89:
1268:
73:
52:
1463:
893:
So there is no injectivity. The true result is that there is a bijection between the ideals of
824:
1570:
1334:
1308:
1200:
719:
700:
449:
1495:
1394:
1130:
1097:
1024:
991:
940:
896:
201:
1167:
798:
356:
318:
292:
1238:
1524:
1443:
1423:
1582:
1546:
1063:
653:
578:
564:
526:
460:
1542:
1160:
are isomorphic if and only if they have the same saturation, or, equivalently, if
1261:, then the localizations are isomorphic (to the zero ring) but the saturation of
637:
I propose the article be renamed 'Localisation (commutative algebra)' or similar
102:
1565:
I'm not an algebraic geometer, so I offer somebody who is one to take a look.
79:
397:
Maximal ideal when localizing versus (the complement of ) a prime ideal
525:
the explanation is a little muddled. I'll take a look at it tonight.
748:
the non-prime intersection of two ideals. Then the localization at
930:
whose all associated prime ideals have an empty intersection with
255:
Thank you for pointing this out: this has now been corrected.
15:
1164:
belongs to one of the multiplicative set, then there exists
1545:. For zero divisors, the second factor must be nonzero.
1265:
is merely the set of zero divisors. Another example is
1527:
1498:
1466:
1446:
1426:
1397:
1369:
are already saturated and different from each other.
1337:
1311:
1271:
1241:
1203:
1170:
1133:
1100:
1027:
994:
943:
899:
856:
827:
801:
758:
722:
359:
321:
295:
503:), but that's not what is needed in this situation.
101:, a collaborative effort to improve the coverage of
341:is nilpotent, then there exists a positive integer
1533:
1513:
1484:
1452:
1432:
1412:
1349:
1323:
1289:
1253:
1227:
1182:
1152:
1119:
1046:
1013:
962:
918:
885:
842:
813:
787:
740:
371:
333:
307:
561:and I don't believe the text implies this at all.
479:The use of the term "annihilator" for the ideal {
519:There isn't technically anything wrong with the
978:whose associated prime ideal do not intersect
459:You are right. I have corrected the sentence.
1082:Isomorphic localization iff same saturation ?
8:
1357:. Then the localizations are isomorphic (to
1222:
1210:
353:is multiplicatively closed, it follows that
289:contains a nilpotent element if and only if
591:You confused ∃ with ∀. The annihilator is {
409:is a prime ideal, then localizing against
47:
1526:
1497:
1465:
1445:
1425:
1396:
1336:
1310:
1305:be the complements of the maximal ideals
1270:
1240:
1202:
1169:
1138:
1132:
1105:
1099:
1032:
1026:
999:
993:
948:
942:
904:
898:
874:
861:
855:
826:
800:
776:
763:
757:
721:
358:
320:
294:
1086:It is stated in the article that : "If
285:No, the property is correct. Note that
49:
19:
505:2A02:810D:980:1704:4446:96B8:E70F:85A8
668:Additions to the 'Properties' section
7:
1391:By definition of the saturation, if
652:I agree. I'll be bold and doing it.
95:This article is within the scope of
1561:Filled in Zariski open sets section
401:The example section says that when
38:It is of interest to the following
1094:are two multiplicative sets, then
14:
1594:Low-priority mathematics articles
1541:divides 0, even when it is not a
667:
523:of the word annihilator here, but
431:? Even when localization is 1-1,
115:Knowledge:WikiProject Mathematics
1197:This appears to be false : take
118:Template:WikiProject Mathematics
82:
72:
51:
20:
1021:has maximal ideals of the form
135:This article has been rated as
435:is not in general an ideal of
279:23:20, 29 September 2011 (UTC)
1:
974:of the primary components of
475:Use of the term "annihilator"
424:. But isn't the ideal really
250:23:23, 7 September 2008 (UTC)
225:This has now been corrected.
210:07:07, 31 December 2008 (UTC)
109:and see a list of open tasks.
1589:C-Class mathematics articles
886:{\displaystyle I_{t}=p_{t}.}
788:{\displaystyle I_{p}=p_{p}.}
587:00:18, 19 October 2014 (UTC)
573:00:11, 19 October 2014 (UTC)
535:20:14, 17 October 2014 (UTC)
513:08:43, 17 October 2014 (UTC)
469:20:24, 14 January 2013 (UTC)
454:19:26, 14 January 2013 (UTC)
190:15:23, 23 October 2008 (UTC)
170:14:57, 18 January 2007 (UTC)
1290:{\displaystyle R=K\times K}
405:is a commutative ring, and
391:00:27, 2 October 2011 (UTC)
1610:
1575:14:24, 17 April 2024 (UTC)
1555:17:49, 20 March 2023 (UTC)
1485:{\displaystyle x\cdot 0=0}
1386:17:20, 20 March 2023 (UTC)
937:The primary components of
843:{\displaystyle t\notin p,}
559:= 0} is rarely an ideal,
1350:{\displaystyle 0\times K}
1324:{\displaystyle K\times 0}
1228:{\displaystyle S=\{0,1\}}
1058:prime and non-maximal in
741:{\displaystyle I=p\cap q}
501:Annihilator (ring theory)
134:
67:
46:
1072:09:10, 30 May 2019 (UTC)
970:are the localization at
687:21:15, 29 May 2019 (UTC)
662:17:19, 24 May 2019 (UTC)
647:16:14, 24 May 2019 (UTC)
621:18:01, 9 June 2015 (UTC)
141:project's priority scale
1514:{\displaystyle x\in R.}
1420:then the saturation of
1413:{\displaystyle 0\in S,}
1194:belongs to the other."
1153:{\displaystyle T^{-1}R}
1120:{\displaystyle S^{-1}R}
1047:{\displaystyle S^{-1}p}
1014:{\displaystyle S^{-1}R}
963:{\displaystyle S^{-1}I}
919:{\displaystyle S^{-1}R}
175:Total ring of fractions
98:WikiProject Mathematics
1535:
1515:
1486:
1454:
1434:
1414:
1351:
1325:
1291:
1255:
1229:
1184:
1183:{\displaystyle t\in R}
1154:
1121:
1048:
1015:
964:
920:
887:
844:
815:
814:{\displaystyle t\in q}
789:
742:
373:
372:{\displaystyle 0\in S}
335:
334:{\displaystyle s\in S}
309:
308:{\displaystyle 0\in S}
28:This article is rated
1536:
1516:
1487:
1455:
1435:
1415:
1352:
1326:
1292:
1256:
1230:
1185:
1155:
1122:
1049:
1016:
965:
921:
888:
845:
816:
790:
743:
442:under the inclusion.
374:
336:
310:
1525:
1496:
1464:
1444:
1424:
1395:
1335:
1309:
1269:
1239:
1201:
1168:
1131:
1098:
1025:
992:
941:
897:
854:
825:
799:
756:
720:
413:yields a local ring
357:
319:
293:
158:Dyadic and etymology
121:mathematics articles
1254:{\displaystyle T=R}
420:with maximal ideal
229:Citation correction
1531:
1511:
1482:
1450:
1430:
1410:
1347:
1321:
1287:
1251:
1225:
1180:
1150:
1117:
1044:
1011:
960:
926:and the ideals of
916:
883:
840:
811:
785:
738:
369:
331:
305:
195:Inverting an ideal
90:Mathematics portal
34:content assessment
1534:{\displaystyle x}
1453:{\displaystyle R}
1433:{\displaystyle S}
269:comment added by
252:
240:comment added by
218:Microlocalization
200:{f^n:n=0,1,...}.
155:
154:
151:
150:
147:
146:
1601:
1540:
1538:
1537:
1532:
1521:In other words,
1520:
1518:
1517:
1512:
1491:
1489:
1488:
1483:
1459:
1457:
1456:
1451:
1439:
1437:
1436:
1431:
1419:
1417:
1416:
1411:
1375:
1368:
1364:
1360:
1356:
1354:
1353:
1348:
1330:
1328:
1327:
1322:
1304:
1300:
1296:
1294:
1293:
1288:
1264:
1260:
1258:
1257:
1252:
1234:
1232:
1231:
1226:
1193:
1189:
1187:
1186:
1181:
1163:
1159:
1157:
1156:
1151:
1146:
1145:
1126:
1124:
1123:
1118:
1113:
1112:
1093:
1089:
1061:
1057:
1053:
1051:
1050:
1045:
1040:
1039:
1020:
1018:
1017:
1012:
1007:
1006:
981:
977:
973:
969:
967:
966:
961:
956:
955:
933:
929:
925:
923:
922:
917:
912:
911:
892:
890:
889:
884:
879:
878:
866:
865:
849:
847:
846:
841:
820:
818:
817:
812:
794:
792:
791:
786:
781:
780:
768:
767:
751:
747:
745:
744:
739:
633:Title of article
388:
383:
378:
376:
375:
370:
340:
338:
337:
332:
315:For example, if
314:
312:
311:
306:
281:
235:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1609:
1608:
1604:
1603:
1602:
1600:
1599:
1598:
1579:
1578:
1563:
1523:
1522:
1494:
1493:
1462:
1461:
1442:
1441:
1422:
1421:
1393:
1392:
1373:
1366:
1362:
1358:
1333:
1332:
1307:
1306:
1302:
1298:
1267:
1266:
1262:
1237:
1236:
1199:
1198:
1191:
1166:
1165:
1161:
1134:
1129:
1128:
1101:
1096:
1095:
1091:
1087:
1084:
1059:
1055:
1028:
1023:
1022:
995:
990:
989:
979:
975:
971:
944:
939:
938:
931:
927:
900:
895:
894:
870:
857:
852:
851:
823:
822:
797:
796:
772:
759:
754:
753:
749:
718:
717:
670:
635:
477:
440:
429:
418:
399:
386:
381:
355:
354:
317:
316:
291:
290:
264:
261:
242:141.154.116.219
231:
220:
197:
177:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1607:
1605:
1597:
1596:
1591:
1581:
1580:
1562:
1559:
1558:
1557:
1530:
1510:
1507:
1504:
1501:
1481:
1478:
1475:
1472:
1469:
1449:
1429:
1409:
1406:
1403:
1400:
1346:
1343:
1340:
1320:
1317:
1314:
1297:, and letting
1286:
1283:
1280:
1277:
1274:
1250:
1247:
1244:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1179:
1176:
1173:
1149:
1144:
1141:
1137:
1116:
1111:
1108:
1104:
1083:
1080:
1079:
1078:
1077:
1076:
1075:
1074:
1043:
1038:
1035:
1031:
1010:
1005:
1002:
998:
985:
984:
983:
959:
954:
951:
947:
915:
910:
907:
903:
882:
877:
873:
869:
864:
860:
839:
836:
833:
830:
810:
807:
804:
795:Similarly, if
784:
779:
775:
771:
766:
762:
737:
734:
731:
728:
725:
709:
708:
696:
669:
666:
665:
664:
634:
631:
630:
629:
628:
627:
626:
625:
624:
623:
476:
473:
472:
471:
438:
427:
416:
398:
395:
394:
393:
368:
365:
362:
349:= 0 and since
330:
327:
324:
304:
301:
298:
271:188.123.231.34
260:
259:Wrong property
257:
230:
227:
219:
216:
214:
196:
193:
176:
173:
167:80.143.125.195
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1606:
1595:
1592:
1590:
1587:
1586:
1584:
1577:
1576:
1572:
1568:
1560:
1556:
1552:
1548:
1544:
1528:
1508:
1505:
1502:
1499:
1479:
1476:
1473:
1470:
1467:
1447:
1427:
1407:
1404:
1401:
1398:
1390:
1389:
1388:
1387:
1383:
1379:
1370:
1344:
1341:
1338:
1318:
1315:
1312:
1284:
1281:
1278:
1275:
1272:
1248:
1245:
1242:
1219:
1216:
1213:
1207:
1204:
1195:
1177:
1174:
1171:
1147:
1142:
1139:
1135:
1114:
1109:
1106:
1102:
1081:
1073:
1069:
1065:
1041:
1036:
1033:
1029:
1008:
1003:
1000:
996:
986:
957:
952:
949:
945:
936:
935:
913:
908:
905:
901:
880:
875:
871:
867:
862:
858:
850:one has also
837:
834:
831:
828:
808:
805:
802:
782:
777:
773:
769:
764:
760:
735:
732:
729:
726:
723:
715:
714:
713:
712:
711:
710:
706:
702:
697:
694:
693:
692:
689:
688:
684:
680:
676:
663:
659:
655:
651:
650:
649:
648:
644:
640:
632:
622:
618:
614:
613:GeoffreyT2000
610:
606:
602:
598:
594:
590:
589:
588:
584:
580:
576:
575:
574:
570:
566:
562:
558:
554:
550:
546:
542:
538:
537:
536:
532:
528:
524:
522:
517:
516:
515:
514:
510:
506:
502:
498:
494:
490:
486:
482:
474:
470:
466:
462:
458:
457:
456:
455:
451:
447:
443:
441:
434:
430:
423:
419:
412:
408:
404:
396:
392:
389:
384:
366:
363:
360:
352:
348:
344:
328:
325:
322:
302:
299:
296:
288:
284:
283:
282:
280:
276:
272:
268:
258:
256:
253:
251:
247:
243:
239:
228:
226:
223:
217:
215:
212:
211:
207:
203:
194:
192:
191:
187:
183:
174:
172:
171:
168:
163:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1564:
1543:zero divisor
1371:
1196:
1085:
691:Two points.
690:
679:Joel Brennan
674:
671:
639:Joel Brennan
636:
608:
604:
600:
596:
592:
560:
556:
552:
548:
544:
540:
539:Also btw, {
520:
518:
496:
492:
488:
484:
480:
478:
444:
436:
432:
425:
421:
414:
410:
406:
402:
400:
350:
346:
342:
286:
265:— Preceding
262:
254:
232:
224:
221:
213:
198:
178:
164:
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
1376:-algebras.
236:—Preceding
202:JackSchmidt
112:Mathematics
103:mathematics
59:Mathematics
1583:Categories
1492:for every
1190:such that
345:such that
182:66.92.4.19
1547:D.Lazard
1460:, since
1064:D.Lazard
675:does not
654:D.Lazard
579:Rschwieb
565:Rschwieb
527:Rschwieb
461:D.Lazard
267:unsigned
238:unsigned
1567:Svennik
446:Thomaso
139:on the
30:C-class
1378:EHecky
1361:) but
752:gives
611:= 0}.
36:scale.
1054:with
1571:talk
1551:talk
1382:talk
1365:and
1331:and
1301:and
1235:and
1127:and
1090:and
1068:talk
821:and
716:Let
705:talk
701:Taku
683:talk
658:talk
643:talk
617:talk
583:talk
569:talk
531:talk
509:talk
465:talk
450:talk
275:talk
246:talk
206:talk
186:talk
1440:is
599:; ∀
547:; ∃
521:use
487:; ∃
411:R-p
379:.--
131:Low
1585::
1573:)
1553:)
1503:∈
1471:⋅
1402:∈
1384:)
1342:×
1316:×
1282:×
1192:st
1175:∈
1140:−
1107:−
1070:)
1062:.
1034:−
1001:−
950:−
906:−
832:∉
806:∈
733:∩
685:)
660:)
645:)
619:)
609:as
607::
603:∈
595:∈
585:)
571:)
557:as
555::
551:∈
543:∈
533:)
511:)
497:as
495::
491:∈
483:∈
467:)
452:)
426:pR
382:PS
364:∈
326:∈
300:∈
277:)
248:)
208:)
188:)
180:--
1569:(
1549:(
1529:x
1509:.
1506:R
1500:x
1480:0
1477:=
1474:0
1468:x
1448:R
1428:S
1408:,
1405:S
1399:0
1380:(
1374:R
1367:T
1363:S
1359:K
1345:K
1339:0
1319:0
1313:K
1303:T
1299:S
1285:K
1279:K
1276:=
1273:R
1263:S
1249:R
1246:=
1243:T
1223:}
1220:1
1217:,
1214:0
1211:{
1208:=
1205:S
1178:R
1172:t
1162:s
1148:R
1143:1
1136:T
1115:R
1110:1
1103:S
1092:T
1088:S
1066:(
1060:R
1056:p
1042:p
1037:1
1030:S
1009:R
1004:1
997:S
982:.
980:S
976:I
972:S
958:I
953:1
946:S
932:S
928:R
914:R
909:1
902:S
881:.
876:t
872:p
868:=
863:t
859:I
838:,
835:p
829:t
809:q
803:t
783:.
778:p
774:p
770:=
765:p
761:I
750:p
736:q
730:p
727:=
724:I
707:)
703:(
681:(
656:(
641:(
615:(
605:S
601:s
597:R
593:a
581:(
567:(
553:S
549:s
545:R
541:a
529:(
507:(
493:S
489:s
485:R
481:a
463:(
448:(
439:p
437:R
433:p
428:p
422:p
417:p
415:R
407:p
403:R
387:T
367:S
361:0
351:S
347:s
343:n
329:S
323:s
303:S
297:0
287:S
273:(
244:(
204:(
184:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
