Endomorphism rings and category equivalences

Garcia, J. G.; Saorı́n, Manuel

doi:10.1016/0021-8693(89)90283-4

Cited by 10 publications

(5 citation statements)

References 19 publications

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“…when M is self-faithful was first shown in [4]. For semisimple or free modules, Corollary 1.3(2) applies, so that MA ≤ c M for each A ≤ c S in these cases.…”

Section: Final Comments the Fact That M Is A Cs-module If And Only Imentioning

confidence: 93%

“…This article began as a search for a conceptual link among these special cases. Eventually, it was realized that a common denominator is the notion of a self-faithful module, a concept first introduced for generators in [6], and recently exploited to good effect in [3], [4] and [5].The principal contribution of this article is to demonstrate that a natural correspondence of closed submodules with closed left ideals occurs whenever M is a self-faithful module. In fact, taking a cue from the approach in [1], we show more generally in Theorem 1.2 that when N is an M -faithful R-module, then there exists an order-preserving correspondence between the closed R-submodules of M and the closed S-submodules of Hom R (M, N ), where S = End R M .…”

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confidence: 99%

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Correspondences of closed submodules

Zelmanowitz

1996

Proc. Amer. Math. Soc.

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Abstract. If N is an M -faithful R-module, then there is an order-preserving correspondence between the closed R-submodules of N and the closed Ssubmodules of Hom R (M, N), where S = End R M.There is a considerable body of research dealing with correspondences between the lattice of submodules of an R-module M and the lattice of left ideals of its endomorphism ring S. This literature includes the well-known Morita theory and its generalizations. When a complete correspondence between all submodules and all left ideals fails to hold, one may still ask whether there is a correspondence between designated sublattices of submodules and left ideals. One particular situation that has attracted much attention has been cases when there is a correspondence between the closed (i.e., essentially closed) submodules of M and the closed left ideals of S. It is known that such a correspondence exists when M is a semisimple module (an elementary observation), when M is a free module [2], when M is a nonsingular retractable module satisfying an additional condition [7], and when M is a nondegenerate module [8]. This article began as a search for a conceptual link among these special cases. Eventually, it was realized that a common denominator is the notion of a self-faithful module, a concept first introduced for generators in [6], and recently exploited to good effect in [3], [4] and [5].The principal contribution of this article is to demonstrate that a natural correspondence of closed submodules with closed left ideals occurs whenever M is a self-faithful module. In fact, taking a cue from the approach in [1], we show more generally in Theorem 1.2 that when N is an M -faithful R-module, then there exists an order-preserving correspondence between the closed R-submodules of M and the closed S-submodules of Hom R (M, N ), where S = End R M . Taking N = M then specializes to the self-faithful case. Additional examples of M being self-faithful, and of the desired correspondence holding, occur when M is a quasi-projective retractable module (Proposition 1.2 in [3]) and when M is a polyform retractable module (Corollary 2.3).Of purely technical interest is the fact that the results contained in this paper remain true even over rings which fail to have an identity element.

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“…when M is self-faithful was first shown in [4]. For semisimple or free modules, Corollary 1.3(2) applies, so that MA ≤ c M for each A ≤ c S in these cases.…”

Section: Final Comments the Fact That M Is A Cs-module If And Only Imentioning

confidence: 93%

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confidence: 99%

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Correspondences of closed submodules

Zelmanowitz

1996

Proc. Amer. Math. Soc.

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“…By applying now [9,Theorem 1.19] to the equivalence U we see that there is a left /?-module M such that R M is a generator, S = E = End( R M), the functor U is given, up to equivalence, by Hom R (M, -) , and, modulo the above isomorphism S = E, the topology %? is the left topology 9 on E given by & = {I C £ £ | £ 0 c / } .…”

Section: (I) There Exists a Generator R M Of I?-mod Such That A Is Ismentioning

confidence: 99%

The characterization problem for endomorphism rings

Garcia

1991

J Aust Math Soc A

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“…Garcia and Saorin [17] call A intrinsically projective if K = Hom^^, KA) for each ls-submodule K of a finitely generated projective right Em odule, and Wisbauer [21] calls the module A an ideal module if the assignment / \-> IA defines a bijection from the set of right ideals of E onto the set of ^-generated submodules of A. These properties are equivalent to (II) when A is a flat left Zs-module, [13], or a Σ-quasi-projective module, [17].…”

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confidence: 99%