22:
1204:
1067:
860:
1059:
760:
1199:{\displaystyle {\begin{array}{rcl}\psi :\mathbb {CFM} _{\mathbb {N} }(R)&\to &\mathbb {CFM} _{\mathbb {N} }(R)^{2}\\M&\mapsto &({\text{odd columns of }}M,{\text{ even columns of }}M)\end{array}}}
1278:
936:
970:
632:
265:. This form reveals that the definition is leftâright symmetric, so it makes no difference whether we define IBN in terms of left or right modules; the two definitions are equivalent.
447:
499:
543:
51:
784:
979:
644:
972:
and with each column having only finitely many non-zero entries. That last requirement allows us to define the product of infinite matrices
371:
Any field satisfies IBN, and this amounts to the fact that finite-dimensional vector spaces have a well defined dimension. Moreover, any
293:
1528:
1499:
73:
343:-modules has a unique rank. The rank is not defined for rings not satisfying IBN. For vector spaces, the rank is also called the
1569:
1559:
1231:
895:
945:
572:
304:), this result is actually equivalent to the definition given here, and can be taken as an alternative definition.
1452:
Abrams, Gene; Ănh, P. N. (2002), "Some ultramatricial algebras which arise as intersections of
Leavitt algebras",
1351:
356:
34:
126:
44:
38:
30:
406:
1554:
452:
289:
55:
1324:
IBN is a necessary (but not sufficient) condition for a ring with no zero divisors to be embeddable in a
1313:
1564:
890:
380:
508:
562:
118:
1344:
1329:
203:
Rephrasing the definition of invariant basis number in terms of matrices, it says that, whenever
138:
95:
550:
1524:
1523:, Graduate Texts in Mathematics, vol. 189, New York: Springer-Verlag, pp. xxiv+557,
1495:
1494:, Graduate Texts in Mathematics, vol. 73, New York: Springer-Verlag, pp. xxiii+502,
1469:
297:
1461:
372:
296:: any two bases for a free module over an IBN ring have the same cardinality. Assuming the
1538:
1509:
1481:
855:{\displaystyle f'\colon \left({\frac {A}{I}}\right)^{n}\to \left({\frac {A}{I}}\right)^{p}}
1534:
1505:
1477:
1054:{\displaystyle \mathbb {CFM} _{\mathbb {N} }(R)\cong \mathbb {CFM} _{\mathbb {N} }(R)^{2}}
344:
301:
384:
1072:
1548:
1340:
1333:
1325:
558:
1210:
122:
292:
number condition is that free modules over an IBN ring satisfy an analogue of the
1214:
347:. Thus the result above is in short: the rank is uniquely defined for all free
320:
110:
91:
87:
1312:. There are other examples of non-IBN rings without this property, among them
1465:
160:
1473:
1384:
1222:
376:
755:{\displaystyle f(i_{1},\dots ,i_{n})=\sum _{k=1}^{n}i_{k}f(e_{k})\in I^{p}}
184:
Equivalently, this means there do not exist distinct positive integers
889:
An example of a nonzero ring that does not satisfy IBN is the ring of
339:. Thus the IBN property asserts that every isomorphism class of free
272:
ring isomorphisms, they are module isomorphisms, even when one of
874:
is an isomorphism between finite dimensional vector spaces, so
352:
15:
1228:
From this isomorphism, it is possible to show (abbreviating
1209:
This infinite matrix ring turns out to be isomorphic to the
976:, giving the ring structure. A left module isomorphism
862:, that can easily be proven to be an isomorphism. Since
121:, the IBN property is the fact that finite-dimensional
1350:
Every ring satisfying the rank condition (i.e. having
1234:
1070:
982:
948:
898:
787:
647:
575:
511:
455:
409:
1273:{\displaystyle \mathbb {CFM} _{\mathbb {N} }(R)=S}
1272:
1198:
1053:
964:
930:
854:
754:
626:
537:
493:
441:
268:Note that the isomorphisms in the definitions are
399:be a commutative ring and assume there exists an
931:{\displaystyle \mathbb {CFM} _{\mathbb {N} }(R)}
43:but its sources remain unclear because it lacks
965:{\displaystyle \mathbb {N} \times \mathbb {N} }
627:{\displaystyle (i_{1},\dots ,i_{n})\in I^{n}}
8:
938:, the matrices with coefficients in a ring
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117:have a well-defined rank. In the case of
74:Learn how and when to remove this message
1332:in the commutative case). See also the
1363:
442:{\displaystyle f\colon A^{n}\to A^{p}}
109:) property if all finitely generated
7:
1354:) must have invariant basis number.
494:{\displaystyle (e_{1},\dots ,e_{n})}
1435:
1371:
294:dimension theorem for vector spaces
148:(IBN) if for all positive integers
288:The main purpose of the invariant
14:
545:is all zeros except a one in the
20:
1490:Hungerford, Thomas W. (1980) ,
300:(a strictly weaker form of the
1304:for any two positive integers
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538:{\displaystyle e_{i}\in A^{n}}
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1:
1521:Lectures on modules and rings
379:) satisfies IBN, as does any
327:of any (and therefore every)
1513:Reprint of the 1974 original
1347:has invariant basis number.
1352:unbounded generating number
1183: even columns of
355:it is uniquely defined for
1586:
1385:"Stacks Project, Tag 0FJ7"
942:, with entries indexed by
1466:10.1142/S0219498802000227
1290:for any positive integer
1389:stacks.math.columbia.edu
29:This article includes a
1519:Lam, Tsit Yuen (1999),
638:-module morphism means
501:the canonical basis of
171:-modules) implies that
58:more precise citations.
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1200:
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891:column finite matrices
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146:invariant basis number
103:invariant basis number
1423:Abrams & Ănh 2002
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509:
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407:
403:-module isomorphism
381:left-Noetherian ring
1570:Homological algebra
1560:Commutative algebra
1438:, Proposition 1.22)
1345:stably finite ring
1330:field of fractions
1270:
1196:
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1051:
962:
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357:finitely generated
31:list of references
1339:Every nontrivial
1184:
1173:
840:
812:
781:-module morphism
549:-th position. By
315:over an IBN ring
311:of a free module
298:ultrafilter lemma
196:is isomorphic to
84:
83:
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1512:
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1454:J. Algebra Appl.
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1314:Leavitt algebras
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373:commutative ring
323:of the exponent
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180:
79:
72:
68:
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54:this article by
45:inline citations
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551:Krull's theorem
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302:axiom of choice
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35:related reading
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1460:(4): 357â363,
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1412:, p. 190)
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385:semilocal ring
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335:isomorphic to
285:
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134:
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125:have a unique
82:
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39:external links
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1555:Module theory
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1532:
1530:0-387-98428-3
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1522:
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1501:0-387-90518-9
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1341:division ring
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1335:
1334:Ore condition
1331:
1327:
1326:division ring
1320:Other results
1319:
1317:
1315:
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1302:
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1293:
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1211:endomorphisms
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1178:
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1061:is given by:
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64:November 2020
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1392:. Retrieved
1388:
1379:
1374:, p. 3)
1366:
1349:
1338:
1323:
1309:
1305:
1300:
1296:
1294:, and hence
1291:
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1282:
1227:
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973:
939:
888:
880:
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871:
870:is a field,
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778:
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375:(except the
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231:matrix over
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215:matrix over
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111:free modules
106:
102:
98:
88:mathematical
85:
70:
61:
50:Please help
42:
1565:Ring theory
1215:free module
1213:of a right
773:induces an
321:cardinality
92:ring theory
56:introducing
1549:Categories
1358:References
363:-modules.
284:Properties
235:such that
192:such that
161:isomorphic
133:Definition
1474:0219-4988
1328:(compare
1223:countable
1163:↦
1113:→
1076:ψ
1013:≅
955:×
825:→
797::
740:∈
691:∑
668:…
612:∈
593:…
523:∈
473:…
427:→
414::
377:zero ring
351:-modules
345:dimension
167:(as left
127:dimension
90:field of
1436:Lam 1999
1372:Lam 1999
793:′
765:because
383:and any
367:Examples
331:-module
101:has the
1539:1653294
1510:0600654
1492:Algebra
1482:1950131
1446:Sources
1394:4 March
1280:) that
561:proper
559:maximal
255:, then
86:In the
52:improve
1537:
1527:
1508:
1498:
1480:
1472:
1225:rank.
553:, let
449:. Let
280:is 1.
223:is an
207:is an
119:fields
1217:over
634:. An
563:ideal
391:Proof
359:free
290:basis
113:over
37:, or
1525:ISBN
1496:ISBN
1470:ISSN
1396:2023
1308:and
569:and
395:Let
309:rank
307:The
245:and
227:-by-
219:and
211:-by-
188:and
152:and
144:has
139:ring
96:ring
94:, a
1462:doi
1343:or
1221:of
565:of
353:iff
276:or
270:not
163:to
107:IBN
1551::
1535:MR
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1387:.
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1299:â
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872:f'
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260:=
250:=
248:BA
240:=
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200:.
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156:,
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1464::
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656:i
652:(
649:f
636:A
620:n
616:I
609:)
604:n
600:i
596:,
590:,
585:1
581:i
577:(
567:A
555:I
547:i
531:n
527:A
518:i
514:e
503:A
489:)
484:n
480:e
476:,
470:,
465:1
461:e
457:(
435:p
431:A
422:n
418:A
411:f
401:A
397:A
361:R
349:R
341:R
337:R
333:R
329:R
325:m
317:R
313:R
278:m
274:n
262:n
258:m
252:I
242:I
233:R
229:m
225:n
221:B
217:R
213:n
209:m
205:A
198:R
194:R
190:n
186:m
178:n
174:m
169:R
165:R
158:R
154:n
150:m
142:R
115:R
105:(
99:R
77:)
71:(
66:)
62:(
48:.
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