  It is shown that, if $R$ is either an Artin algebra or a commutative
noetherian domain of Krull dimension $1$, then infinite direct products of
$R$-modules resist direct sum decomposition as follows: If $(M_n)_{n \in \Bbb
N}$ is a family of non-isomorphic, finitely generated, indecomposable
$R$-modules, then $\prod_{n\in \Bbb N} M_n$ is not a direct sum of finitely
generated modules. The bearing of this direct product condition on the pure
semisimplicity problem is discussed.
