184:
292:
131:
65:
in von
Neumann algebras can be extended to all Baer *-rings, For example, Baer *-rings can be divided into types I, II, and III in the same way as von Neumann algebras.
476:
446:
549:
who studied a similar property in operator algebras. This "principal implies projective" condition is the reason
Rickart rings are sometimes called PP-rings. (
251:. For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric.
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is a *-ring such that left annihilator of any element is generated (as a left ideal) by a projection.
260:
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is a *-ring such that left annihilator of any subset is generated (as a left ideal) by a projection.
381:
62:
47:
21:
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110:
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642:, Die Grundlehren der mathematischen Wissenschaften, vol. 195, Berlin, New York:
854:
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are a Baer ring and is also a Baer *-ring with the involution * given by the adjoint.
494:
299:
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255:
In operator theory, the definitions are strengthened slightly by requiring the ring
384:
are projective, it is clear that both types are left
Rickart rings. This includes
322:
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637:
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42:
363:
204:(For unital rings) the left annihilator of any element is a direct summand of
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von
Neumann algebras are examples of all the different sorts of ring above.
388:, which are left and right semihereditary. If a von Neumann regular ring
782:
702:
314:
774:
686:
68:
In the literature, left
Rickart rings have also been termed left
305:, the definition of Rickart *-ring is left-right symmetric.
732:, Graduate Texts in Mathematics No. 189, Berlin, New York:
72:. ("Principal implies projective": See definitions below.)
243:(For unital rings) The left annihilator of any subset of
240:
is generated (as a left ideal) by an idempotent element.
201:
is generated (as a left ideal) by an idempotent element.
46:
are various attempts to give an algebraic analogue of
761:(1946), "Banach algebras with an adjoint operation",
458:
428:
268:
192:
is a ring satisfying any of the following conditions:
139:
113:
470:
440:
286:
178:
125:
579:T.Y. Lam (1999), "Lectures on Modules and Rings"
211:All principal left ideals (ideals of the form
197:the left annihilator of any single element of
8:
799:"Regular ring (in the sense of von Neumann)"
465:
459:
435:
429:
173:
170:
164:
140:
509:The projections in a Rickart *-ring form a
671:(1951), "Projections in Banach algebras",
448:except for the annihilator of 0, which is
376:Since the principal left ideals of a left
457:
427:
359:
267:
138:
112:
546:
538:
179:{\displaystyle \{r\in R\mid rX=\{0\}\}}
607:Linear algebra and projective geometry
411:left and right ideals are summands in
236:The left annihilator of any subset of
7:
567:
422:is Baer, since all annihilators are
550:
61:*-ring, and much of the theory of
14:
723:, New York: W. A. Benjamin, Inc.
287:{\displaystyle *:R\rightarrow R}
636:Berberian, Sterling K. (1972),
562:This condition was studied by
545:Rickart rings are named after
517:if the ring is a Baer *-ring.
278:
231:has the following definitions:
1:
730:Lectures on modules and rings
415:, including the annihilators.
57:Any von Neumann algebra is a
126:{\displaystyle X\subseteq R}
89:which has the property that
840:Encyclopedia of Mathematics
822:Encyclopedia of Mathematics
804:Encyclopedia of Mathematics
366:that is also a Baer *-ring.
882:
815:L.A. Skornyakov (2001) ,
797:L.A. Skornyakov (2001) ,
386:von Neumann regular rings
491:bounded linear operators
85:of a ring is an element
833:J.D.M. Wright (2001) ,
728:Lam, Tsit-Yuen (1999),
247:is a direct summand of
717:Kaplansky, I. (1968),
472:
442:
392:is also right or left
288:
180:
127:
763:Annals of Mathematics
674:Annals of Mathematics
473:
471:{\displaystyle \{0\}}
443:
441:{\displaystyle \{0\}}
289:
181:
128:
50:, using axioms about
861:Von Neumann algebras
456:
426:
294:. Since this makes
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137:
111:
48:von Neumann algebras
382:semihereditary ring
190:(left) Rickart ring
22:functional analysis
720:Rings of Operators
468:
438:
298:isomorphic to its
284:
176:
123:
83:idempotent element
765:, Second Series,
743:978-0-387-98428-5
677:, Second Series,
669:Kaplansky, Irving
653:978-3-540-05751-2
621:978-0-486-44565-6
564:Reinhold Baer
317:is an idempotent
54:of various sets.
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847:
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560:
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527:Baer *-semigroup
482:are summands of
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360:Kaplansky (1951)
358:, introduced by
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18:abstract algebra
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775:10.2307/1969091
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734:Springer-Verlag
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687:10.2307/1969540
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644:Springer-Verlag
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407:is Baer, since
405:semisimple ring
378:hereditary ring
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38:Rickart *-rings
12:
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5:
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863:
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849:
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817:"Rickart ring"
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794:
769:(3): 528–550,
759:Rickart, C. E.
755:
742:
725:
714:
681:(2): 235–249,
665:
652:
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612:Academic Press
610:, Boston, MA:
602:Baer, Reinhold
596:
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547:Rickart (1946)
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394:self injective
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340:Rickart *-ring
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836:
835:"AW* algebra"
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585:0-387-98428-3
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495:Hilbert space
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300:opposite ring
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35:
34:Rickart rings
31:
27:
23:
19:
838:
820:
802:
766:
762:
729:
719:
678:
672:
639:Baer *-rings
638:
606:
575:
558:
541:
508:
489:The ring of
483:
479:
449:
412:
408:
397:
389:
353:
346:
339:
331:
327:
323:self-adjoint
318:
310:
302:
295:
256:
254:
248:
244:
237:
228:
219:
212:
205:
198:
189:
101:
94:
90:
86:
69:
67:
56:
52:annihilators
43:AW*-algebras
41:
37:
33:
30:Baer *-rings
29:
25:
15:
866:Ring theory
513:, which is
452:, and both
355:AW*-algebra
347:Baer *-ring
259:to have an
104:annihilator
76:Definitions
63:projections
855:Categories
595:References
505:Properties
364:C*-algebra
311:projection
261:involution
217:projective
26:Baer rings
845:EMS Press
827:EMS Press
809:EMS Press
695:0003-486X
279:→
270:∗
229:Baer ring
153:∣
147:∈
118:⊆
107:of a set
604:(1952),
551:Lam 1999
521:See also
515:complete
400:is Baer.
380:or left
371:Examples
321:that is
222:modules.
70:PP-rings
791:0017474
783:1969091
752:1653294
711:0042067
703:1969540
662:0429975
630:0052795
566: (
511:lattice
396:, then
362:, is a
789:
781:
750:
740:
709:
701:
693:
660:
650:
628:
618:
587:pp.260
583:
420:domain
315:*-ring
215:) are
40:, and
779:JSTOR
699:JSTOR
533:Notes
493:on a
313:in a
102:left
738:ISBN
691:ISSN
648:ISBN
616:ISBN
581:ISBN
568:1952
478:and
418:Any
403:Any
330:* =
100:The
59:Baer
20:and
771:doi
683:doi
409:all
352:An
133:is
81:An
24:,
16:In
857::
843:,
837:,
825:,
819:,
807:,
801:,
787:MR
785:,
777:,
767:47
748:MR
746:,
736:,
707:MR
705:,
697:,
689:,
679:53
658:MR
656:,
646:,
626:MR
624:,
614:,
570:).
345:A
338:A
335:).
309:A
227:A
213:Rx
188:A
93:=
36:,
32:,
28:,
773::
685::
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486:.
484:R
480:R
466:}
463:0
460:{
450:R
436:}
433:0
430:{
413:R
398:R
390:R
332:p
328:p
325:(
319:p
303:R
296:R
282:R
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273::
257:R
249:R
245:R
238:R
220:R
208:.
206:R
199:R
174:}
171:}
168:0
165:{
162:=
159:X
156:r
150:R
144:r
141:{
121:R
115:X
97:.
95:e
91:e
87:e
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