Direct sum decompositions of infinitely generated modules
D. J. Benson
Wayne W. Wheeler

\begin{abstract}
Almost all of the basic theorems in the representation theory of
finite groups have proofs that depend upon the Krull--Schmidt
Theorem.  Because this theorem holds only for finite-dimensional
modules, however, the recent interest in infinitely generated modules
raises the question of which results may hold more generally.  In this
paper we present an example showing that Green's Indecomposability
Theorem fails for infinitely generated modules.  By developing and
applying some general properties of idempotent modules, we are also
able to construct explicit examples of modules for which the
cancellation property fails.
\end{abstract}