Direct sums of representations as modules over their endomorphism rings

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 = $\text{End}$R \bigl( \bigoplus{i \in I} M_i \bigr)$,howmuchpressure,intermsofthe$S$-structureof$\bigoplus_{i \in I} M_i$,isrequiredtoforcethe$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 $\bigoplus_{i \in I} M_i$isendo-Artinianifandonlyifitis$\Sigma$-algebraically compact.