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Endomorphism rings of essentially pseudo injetive modules. (English) Zbl 1309.16003

Summary: Some results on the endomorphism rings of essentially pseudo injective modules are obtained. In particular, it is proved that for a uniform essentially pseudo injective module \(M\), the socle of \(M\) is essential in \(M\) iff Jacobson radical of endomorphism ring of \(M\) is equal to the set of all homomorphisms from socle of \(M\) to \(M\). It is shown that the endomorphism ring of an essentially pseudo injective uniform module is local and the mono-endomorphism of an essentially pseudo injective uniform module is an automorphism. Finally, we find a characterization for a uniform module \(M\) to be essentially pseudo injective in terms of its injective hull.

MSC:

16D50 Injective modules, self-injective associative rings
16S50 Endomorphism rings; matrix rings
16W20 Automorphisms and endomorphisms
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