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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)

16D90 Module categories in associative algebras
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