We study the category $\operatorname{ Morph}(\operatorname{ Mod} R)$ whose objects are all morphisms between two right $R$-modules. The behavior of objects of $\operatorname{ Morph}(\operatorname{ Mod} R)$ whose endomorphism ring in $\operatorname{ Morph}(\operatorname{ Mod} R)$ is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum $\oplus_{ i=1}^nM_i$, that is, block-diagonal decompositions, where each object $M_i$ of $\operatorname{ Morph}(\operatorname{ Mod} R)$ 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 $\operatorname{ End}(M_{ j,i})$, 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 $\oplus_{ i=1}^nM_i$ depend on four invariants.
Categories: math.RA