Homomorphisms with Semilocal Endomorphism Rings Between Modules

We study the category Morph(Mod- R ) whose objects are all morphisms between two right R -modules. The behavior of the objects of Mod- R whose endomorphism ring in Morph(Mod- R ) is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum ⊕ i = 1 n M i , that is, block-diagonal decompositions, where each object M i of Morph(Mod- R ) denotes a morphism μ M i : M 0 , i → M 1 , i and where all the modules M j , i have a local endomorphism ring End( M j , i ), depend on two invariants. This behavior is very similar to that of direct-sum decompositions of serial modules of finite Goldie dimension, which also depend on two invariants (monogeny class and epigeny class). When all the modules M j , i are uniserial modules, the direct-sum decompositions (block-diagonal decompositions) of a direct-sum ⊕ i = 1 n M i depend on four invariants.