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=1nMi, that is, block-diagonal decompositions, where each object Mi of Morph(Mod-R) denotes a morphism μMi:M0,i→M1,i and where all the modules Mj,i have a local endomorphism ring End(Mj,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 Mj,i are uniserial modules, the direct-sum decompositions (block-diagonal decompositions) of a direct-sum ⊕i=1nMi depend on four invariants.