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This paper is devoted to the study of the endo-structure of infinite directsums $\bigoplus_{i \in I} M_i$ 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 $M_i$ intofinitely many isomorphism classes? In case the $M_i$ are endofinite (i.e., offinite length over their endomorphism rings), the number of isomorphism classesamong the $M_i$ is finite if and only if $\bigoplus_{i \in I} M_i$ isendo-noetherian and the $M_i$ form a right $T$-nilpotent class. This is acorollary of a more general theorem in the paper which features the weakerconditions of (right or left) semi-$T$-nilpotence as well as the endosocle of amodule. This result is sharpened in the case of Artin algebras, by showing thatthen, 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.

Direct sums of representations as modules over their endomorphism rings