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Properties of endomorphism rings of modules and their duals. (English) Zbl 0588.16025

Let \({}_ RM\) be a left R-module, \(B=End_ RM\) and \(M^*=Hom_ R(M,R)\) be its dual module. The author investigates some left-right properties of B which are symmetrically present on M and \(M^*_ R\). He proves that B is left strongly modular if and only if any element of B which has zero kernel in \({}_ RM\) has essential image in \({}_ RM\), while B is right strongly modular if and only if any element of B which has zero kernel in \(M^*_ R\) has essential image in \(M^*_ R\). The ring B is a left Utumi ring if and only if every submodule \({}_ RU\) of \({}_ RM\) such that \(U^{\perp}=0\) is essential in M, while B is right Utumi ring if and only if every submodule \(U^*_ R\) of \(M^*_ R\) such that \(^{\perp}U^*=0\) is essential in \(M^*_ R\).
Reviewer: L.Ioffe

MSC:

16S50 Endomorphism rings; matrix rings
16P50 Localization and associative Noetherian rings
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References:

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