Abstract
Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring. Any semi simple ring does not satisfy this property. We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.
Highlights
Let M, N be modules over a ring R and set of all R-homomorphisms from M to N is written HomR(M, N )
Let R be the upper triangular ring over a division ring and F be the category of R-modules which have a summand isomorphic to R
Let M be a module over a ring R and e an idempotent in EndR(M )
Summary
Let M, N be modules over a ring R and set of all R-homomorphisms from M to N is written HomR(M, N ). EndR(M ) = HomR(M, M ) is a ring over addition and composition of mapping, and called endomorphism ring of M . In general if two modules are isomorphic their endomorphism rings are isomorphic. The Baer-Kaplansky theorem states that two Abelian torsion groups are isomorphic if and only if their endomorphism rings are isomorphic (see [2] and [4]).
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