Homomorphisms with semilocal endomorphism rings between modules
Federico Campanini, Susan F. El-Deken, Alberto Facchini
TL;DR
This paper extends block-decomposition theory to the morphism category Morph(Mod-R) by focusing on objects with semilocal endomorphism rings. It develops a canonical framework of functors ($D$, $C$, ${ m Ker}$, ${ m Coker}$, and $U$) and a monoid $V(M)$ to track decompositions, linking Morph(Mod-R) to modules over a triangular matrix ring $T$. It proves semilocality and finiteness-type results for endomorphism rings $E_M$ and analyzes decompositions via domain/codomain invariants, Krull monoid structures, and Krull-Schmidt-type theorems, including specialized results for morphisms between uniserial modules. The work yields a rich invariant-based classification of block decompositions, with implications for direct-sum theory in module categories and a parallel to classical invariants like lower part/epigeny class in matrix decompositions.
Abstract
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 $μ_{M_i}\colon M_{0,i}\to M_{1,i}$ 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.