Homomorphisms with Semilocal Endomorphism Rings Between Modules

Abstract

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 $\oplus _{i=1}^{n}M_{i}$, that is, block-diagonal decompositions, where each object M_i of Morph(Mod-R) denotes a morphism $\mu _{M_{i}}\colon M_{0,i}\to M_{1,i}$ 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 $\oplus _{i=1}^{n}M_{i}$ depend on four invariants.