Direct sums of representations as modules over their endomorphism rings

Birge Huisgen-Zimmermann; Manuel Saorín

paper

Direct sums of representations as modules over their endomorphism rings

arXiv:1407.2364

Abstract

This paper is devoted to the study of the endo-structure of infinite direct sums $⨁_{i \in I} M_{i}$ of indecomposable modules $M_{i}$ over a ring $R$ . It is centered on the following question: If $S = End_{R} (⨁_{i \in I} M_{i})$ , how much pressure, in terms of the $S$ -structure of $⨁_{i \in I} M_{i}$ , 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 $⨁_{i \in I} M_{i}$ 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 $⨁_{i \in I} M_{i}$ is endo-Artinian if and only if it is $Σ$ -algebraically compact.