Direct sums of representations as modules over their endomorphism rings (Q1602012)

A module is endofinite if it has finite length when regarded as a module over its endomorphism ring. If \(M\) is endofinite then every direct summand of any product of copies of \(M\) also is endofinite, indeed all its indecomposable summands occur already as direct summands of \(M\). A direct sum of infinitely many different endofinite modules need not, however, be endofinite (think of direct sums of finite Abelian groups for example). The authors consider the following question: if \(S=\text{End}(M)\) where \(M=\bigoplus_iM_i\) with the \(M_i\) endofinite ``how much pressure, in terms of the \(S\)-structure of \(\bigoplus_iM_i\), is required to force the \(M_i\) into finitely many isomorphism classes?'' They provide an answer: exactly if \(\bigoplus_iM_i\) is endo-Noetherian and the \(M_i\) form a right \(T\)-nilpotent class. This is a corollary of a more general theorem in their paper which features the weaker conditions of (right or left) semi-\(T\)-nilpotence as well as the endosocle of a module. The paper contains a number of interesting results relating to the above question.