475:
is algebraically compact. This, together with the fact that all finite modules are algebraically compact, gives rise to the intuition that algebraically compact modules are those (possibly "large") modules which share the nice properties of "small" modules.
285:
if, for all such systems, if every subsystem formed by a finite number of the equations has a solution, then the whole system has a solution. (The solutions to the various subsystems may be different.)
35:
that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of
149:
43:, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.
421:
239:
196:
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Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of
762:
608:. One can often simplify a problem by first applying the *-functor, since algebraically compact modules are easier to deal with.
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472:
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such that all the equations of the system are simultaneously satisfied. (It is not required that only finitely many
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Module such that infinite systems of linear equations can be solved by solving finite subsystems
758:
688:
573:
442:
40:
757:. London Mathematical Society Lecture Note Series: Cambridge University Press, Cambridge.
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480:
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It turns out that a module is algebraically compact if and only if it is pure-injective.
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787:
492:
484:
438:
592:* is algebraically compact. Furthermore, there are pure injective homomorphisms
20:
685:
441:
is algebraically compact (since it is pure-injective). More generally, every
328:
522:, this is a complete list of indecomposable algebraically compact modules.
502:
is algebraically compact as both a module over itself and a module over
577:
434:
All modules with finitely many elements are algebraically compact.
73:-module. Consider a system of infinitely many linear equations
525:
Many algebraically compact modules can be produced using the
709:-modules precisely correspond to the injective objects in
39:. These algebraically compact modules are analogous to
390:
208:
168:
82:
415:
244:The goal is to decide whether such a system has a
233:
190:
143:
144:{\displaystyle \sum _{j\in J}r_{i,j}x_{j}=m_{i},}
445:is algebraically compact, for the same reason.
548:, one forms the (algebraic) character module
8:
636:can be extended to a module homomorphism
616:The following condition is equivalent to
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213:
207:
173:
167:
132:
119:
103:
87:
81:
306:if the induced homomorphism between the
745:
248:, that is whether there exist elements
705:under which the algebraically compact
572:-module, and the *-operation yields a
7:
588:-modules. Every module of the form
684:algebraically compact module has a
353:if any pure injective homomorphism
14:
728:-module and to a direct sum of
510:are algebraically compact as a
416:{\displaystyle f\circ j=1_{M}}
1:
620:being algebraically compact:
25:algebraically compact modules
776:C.U. Jensen and H. Lenzing:
724:to an algebraically compact
234:{\displaystyle r_{i,j}\in R}
514:-module. Together with the
191:{\displaystyle m_{i}\in M,}
815:
672:, one for each element of
656:, one for each element of
483:are algebraically compact
780:, Gordon and Breach, 1989
755:Model theory and modules
778:Model Theoretic Algebra
491:-modules). The ring of
370:(that is, there exists
732:algebraically compact
568:. This is then a left
536:of abelian groups. If
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235:
202:the number of nonzero
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29:pure-injective modules
722:elementary equivalent
700:Grothendieck category
544:module over the ring
527:injective cogenerator
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289:On the other hand, a
283:algebraically compact
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753:Prest, Mike (1988).
624:For every index set
552:* consisting of all
518:finite modules over
467:-module with finite
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166:
80:
37:module homomorphisms
628:, the addition map
554:group homomorphisms
454:associative algebra
291:module homomorphism
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188:
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98:
689:endomorphism ring
584:-modules to left
456:with 1 over some
162:may be infinite,
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41:injective modules
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508:rational numbers
443:injective module
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331:for every right
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154:where both sets
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498:for each prime
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308:tensor products
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274:are non-zero.)
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730:indecomposable
682:indecomposable
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576:contravariant
516:indecomposable
496:-adic integers
485:abelian groups
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351:pure-injective
304:pure embedding
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27:, also called
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794:Module theory
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764:0-521-34833-1
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481:Prüfer groups
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463:, then every
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343:. The module
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284:
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198:and for each
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799:Model theory
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698:-Mod into a
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669:
664:denotes the
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648:denotes the
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439:vector space
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24:
18:
720:-module is
580:from right
277:The module
241:is finite.
47:Definitions
21:mathematics
788:Categories
740:References
736:-modules.
650:direct sum
473:dimension
395:∘
329:injective
226:∈
180:∈
92:∈
85:∑
574:faithful
430:Examples
375: :
358: :
337:-module
246:solution
666:product
602:natural
578:functor
67:a left
33:modules
761:
716:Every
680:Every
644:(here
506:. The
487:(i.e.
452:is an
437:Every
368:splits
61:, and
31:, are
686:local
612:Facts
556:from
542:right
540:is a
458:field
384:with
302:is a
57:be a
759:ISBN
600:**,
479:The
158:and
59:ring
51:Let
604:in
560:to
448:If
423:).
349:is
327:is
281:is
259:of
19:In
790::
713:.
691:.
676:).
660:;
640:→
632:→
596:→
379:→
362:→
322:⊗
318:→
314:⊗
297:→
23:,
767:.
734:R
726:R
718:R
711:G
707:R
703:G
696:R
674:I
670:M
662:M
658:I
654:M
646:M
642:M
638:M
634:M
630:M
626:I
618:M
606:H
598:H
594:H
590:H
586:R
582:R
570:R
566:Z
564:/
562:Q
558:H
550:H
546:R
538:H
534:Z
532:/
530:Q
520:Z
512:Z
504:Z
500:p
494:p
489:Z
471:-
469:k
465:R
461:k
450:R
409:M
405:1
401:=
398:j
392:f
381:M
377:K
373:f
364:K
360:M
356:j
346:M
340:C
334:R
324:K
320:C
316:M
312:C
299:K
295:M
279:M
270:j
268:x
262:M
255:j
251:x
229:R
221:j
218:,
215:i
211:r
200:i
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183:M
175:i
171:m
160:J
156:I
139:,
134:i
130:m
126:=
121:j
117:x
111:j
108:,
105:i
101:r
95:J
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70:R
64:M
54:R
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