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On the structure of *-modules. (English) Zbl 0795.16005
Let $$R$$ be an associative ring with identity. For a right $$R$$-module $$P_ R$$ and an arbitrary cogenerator $$Q_ R$$ of $$\text{Mod-}R$$ we set $$K_ A= \operatorname{Hom}_ R(P,Q)$$, where $$A= \text{End}(P_ R)$$ is the endomorphism ring of $$P$$. Recall that the module $$P_ R$$ is called a $$*$$-module if the functors $$-\otimes_ A P$$ and $$\operatorname{Hom}_ R(P,-)$$ define the category equivalence between $$\text{Cogen} (K_ A)$$ and $$\text{Gen}(P_ R)$$. In the second paragraph the authors prove that over a commutative ring $$R$$ every finitely generated $$*$$-module $$P_ R$$ is a quasi-progenerator (Theorem 2.4). If, moreover, $$P_ R$$ is faithful, then it is a progenerator (Theorem 2.3). A different situation arises in the case of modules over non-commutative rings. The main result in the third part characterizes the rings $$R$$ possessing a $$*$$-module $$P_ R$$ such that $$\text{Gen}(P_ R)$$ coincides with the case of all modules as right hereditary Noetherian rings such that $$E(R_ R)$$ (the minimal injective cogenerator of $$\text{Mod-}R$$) is finitely generated.
Reviewer: L.Bican (Praha)

MSC:
 16D90 Module categories in associative algebras
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