Direct products of modules and the pure semisimplicity conjecture

Abstract

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 $(M_n{)}_{n\in \mathbb{N}}$ is a family of non-isomorphic, finitely generated, indecomposable $R$-modules, then ${\prod }_{n\in \mathbb{N}}{M}_n$ is not a direct sum of finitely generated modules. The bearing of this direct product condition on the pure semisimplicity problem is discussed.