  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.
