Direct sums of representations as modules over their endomorphism rings

TL;DR

This paper investigates the endo-structure of infinite direct sums of indecomposable modules over a ring, establishing conditions under which the modules fall into finitely many isomorphism classes based on their endomorphism ring properties.

Contribution

It provides new criteria linking endo-structure conditions like endo-noetherianity and T-nilpotence to the finiteness of isomorphism classes among summands, especially for endofinite modules.

Findings

Finite isomorphism classes occur if the sum is endo-noetherian and T-nilpotent.

For Artin algebras, endo-Artinian sums are equivalent to being $\\Sigma$-algebraically compact.

General conditions involve semi-T-nilpotence and endosocle properties.

Abstract

This paper is devoted to the study of the endo-structure of infinite direct sums $\bigoplus_{i \in I} M_i$ of indecomposable modules $M_i$ over a ring $R$. It is centered on the following question: If $S = \text{End}_R \bigl( \bigoplus_{i \in I} M_i \bigr)$, how much pressure, in terms of the $S$-structure of $\bigoplus_{i \in I} M_i$, is required to force the $M_i$ into finitely many isomorphism classes? In case the $M_i$ are endofinite (i.e., of finite length over their endomorphism rings), the number of isomorphism classes among the $M_i$ is finite if and only if $\bigoplus_{i \in I} M_i$ is endo-noetherian and the $M_i$ form a right $T$-nilpotent class. This is a corollary of a more general theorem in the paper which features the weaker conditions of (right or left) semi-$T$-nilpotence as well as the endosocle of a module. This result is sharpened in the case of Artin algebras, by showing that then, if the $M_i$ are finitely generated, the direct sum $\bigoplus_{i \in I} M_i$ is endo-Artinian if and only if it is $\Sigma$-algebraically compact.