894:
139:
497:
416:
609:
293:
680:
195:
344:
of interest, the idealizer is defined more simply by taking advantage of the fact that multiplication by ring elements is already absorbed on one side. Explicitly,
53:
308:, however it is apparent that this set is actually the Lie ring equivalent of the idealizer. It is not necessary to specify that ⊆
935:
431:
350:
842:
549:
837:, Mathematical Surveys and Monographs, vol. 174, Providence, RI: American Mathematical Society, pp. iv+228,
868:
928:
242:
629:
969:
162:
921:
893:
954:
44:
959:
134:{\displaystyle \mathbb {I} _{S}(T)=\{s\in S\mid sT\subseteq T{\text{ and }}Ts\subseteq T\}.}
964:
878:
852:
826:
526:
8:
821:, Pure and Applied Mathematics, No. 33, New York: Marcel Dekker Inc., pp. viii+206,
510:
715:
699:
153:
864:
838:
313:
513:, the idealizer is related to a more general construction. Given a commutative ring
909:
17:
874:
848:
822:
905:
726:
948:
753:
901:
225:
213:
145:
733:
300:
29:
221:
492:{\displaystyle \mathbb {I} _{R}(L)=\{r\in R\mid Lr\subseteq L\}}
411:{\displaystyle \mathbb {I} _{R}(T)=\{r\in R\mid rT\subseteq T\}}
340:
Often, when right or left ideals are the additive subgroups of
863:, Dordrecht: Kluwer Academic Publishers, pp. xvi+618,
770:
615:
In terms of this conductor notation, an additive subgroup
794:
782:
859:
Mikhalev, Alexander V.; Pilz, Günter F., eds. (2002),
632:
552:
434:
353:
316:
of the Lie product causes = − ∈
245:
165:
56:
604:{\displaystyle (A:B):=\{r\in R\mid Br\subseteq A\}}
674:
603:
491:
410:
287:
189:
133:
946:
835:Hereditary Noetherian prime rings and idealizers
748:is any faithful nondegenerate representation of
698:, the conductor is part of the structure of the
929:
858:
833:Levy, Lawrence S.; Robson, J. Chris (2011),
776:
598:
571:
486:
459:
405:
378:
282:
246:
197:(defined in the multiplicative semigroup of
125:
81:
936:
922:
832:
819:Ring theory: Nonsingular rings and modules
800:
288:{\displaystyle \{r\in L\mid \subseteq S\}}
675:{\displaystyle \mathbb {I} _{R}(B)=(B:B)}
635:
437:
356:
168:
59:
816:
788:
947:
888:
190:{\displaystyle \mathbb {I} _{R}(A)}
13:
14:
981:
892:
47:. Such an idealizer is given by
861:The concise handbook of algebra
35:is the largest subsemigroup of
669:
657:
651:
645:
565:
553:
453:
447:
372:
366:
273:
261:
184:
178:
75:
69:
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810:
152:is an additive subgroup of a
908:. You can help Knowledge by
201:) is the largest subring of
7:
335:
232:is an additive subgroup of
10:
986:
887:
324:is the largest subring of
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298:is classically called the
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817:Goodearl, K. R. (1976),
777:Mikhalev & Pilz 2002
763:
517:, and given two subsets
904:-related article is a
801:Levy & Robson 2011
676:
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493:
412:
289:
209:is a two-sided ideal.
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677:
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425:is a right ideal, or
413:
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736:to the idealizer of
630:
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511:commutative algebra
970:Group theory stubs
716:multiplier algebra
700:residuated lattice
672:
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285:
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24:of a subsemigroup
917:
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844:978-0-8218-5350-4
506:is a left ideal.
314:anticommutativity
111:
977:
955:Abstract algebra
938:
931:
924:
896:
889:
881:
855:
829:
804:
798:
792:
786:
780:
774:
681:
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498:
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495:
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365:
364:
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332:is a Lie ideal.
294:
292:
291:
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196:
194:
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171:
140:
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132:
112:
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18:abstract algebra
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984:
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979:
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623:has idealizer
551:
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435:
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348:
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240:
236:, then the set
172:
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600:
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9:
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3:
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982:
971:
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963:
961:
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950:
939:
934:
932:
927:
925:
920:
919:
913:
911:
907:
903:
898:
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891:
890:
886:
880:
876:
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870:0-7923-7072-4
866:
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850:
846:
840:
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820:
815:
814:
802:
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790:
789:Goodearl 1976
785:
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769:
761:
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754:Hilbert space
751:
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739:
735:
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728:
724:
720:
717:
709:
708:
707:
705:
702:of ideals of
701:
697:
693:
689:
666:
663:
660:
654:
648:
640:
626:
625:
624:
622:
618:
595:
592:
589:
586:
583:
580:
577:
574:
568:
562:
559:
556:
546:
545:
544:
543:is given by
542:
538:
534:
531:
529:
524:
520:
516:
512:
507:
505:
483:
480:
477:
474:
471:
468:
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462:
456:
450:
442:
428:
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426:
424:
402:
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375:
369:
361:
347:
346:
345:
343:
333:
331:
327:
323:
319:
315:
311:
307:
303:
302:
279:
276:
270:
267:
264:
258:
255:
252:
249:
239:
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237:
235:
231:
227:
223:
219:
215:
210:
208:
204:
200:
181:
173:
158:
155:
151:
147:
128:
122:
119:
116:
113:
105:
102:
99:
96:
93:
90:
87:
84:
78:
72:
64:
50:
49:
48:
46:
42:
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34:
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960:Group theory
910:expanding it
902:group theory
899:
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749:
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741:
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149:
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25:
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965:Ring theory
541:transporter
525:of a right
226:Lie algebra
214:Lie algebra
146:ring theory
949:Categories
811:References
734:isomorphic
727:C*-algebra
312:, because
301:normalizer
593:⊆
584:∣
578:∈
537:conductor
481:⊆
472:∣
466:∈
400:⊆
391:∣
385:∈
328:in which
277:⊆
259:∣
253:∈
205:in which
120:⊆
103:⊆
94:∣
88:∈
39:in which
30:semigroup
22:idealizer
791:, p.121.
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336:Comments
222:Lie ring
879:1966155
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867:
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803:, p.7.
756:
746:π
738:π
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686:When
220:is a
216:, if
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