Let A and B be rings, A a subcategory of Mod-A closed under submodules and containing AA, and B a subcategory of Mod-B closed under direct sums and epimorphic images. Then any equivalence between A and B is represented by a bimodule APB with A=EndPB via the functors --⊗AP and HomB(P,–). This was proved by Menini and A. Orsatti [Rend. Sem. Mat. Univ. Padova 82 (1989), 203--231 (1990); MR1049594 (91h:16026)], and such a module is called a ∗-module. For instance, every quasi-progenerator and every tilting module is a ∗-module. The purpose of this paper is, in the authors' words, "to measure the gaps between the classes of ∗-modules, of quasi-progenerators and of tilting modules''. Among the results in this direction, it is shown that every finitely generated ∗-module over a commutative ring is a quasi-progenerator; and that this is not the case in general.
On the structure of $*$-modules
COLPI, RICCARDO;
1993
Abstract
Let A and B be rings, A a subcategory of Mod-A closed under submodules and containing AA, and B a subcategory of Mod-B closed under direct sums and epimorphic images. Then any equivalence between A and B is represented by a bimodule APB with A=EndPB via the functors --⊗AP and HomB(P,–). This was proved by Menini and A. Orsatti [Rend. Sem. Mat. Univ. Padova 82 (1989), 203--231 (1990); MR1049594 (91h:16026)], and such a module is called a ∗-module. For instance, every quasi-progenerator and every tilting module is a ∗-module. The purpose of this paper is, in the authors' words, "to measure the gaps between the classes of ∗-modules, of quasi-progenerators and of tilting modules''. Among the results in this direction, it is shown that every finitely generated ∗-module over a commutative ring is a quasi-progenerator; and that this is not the case in general.Pubblicazioni consigliate
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