%0 Journal Article %T Direct products of modules and the pure semisimplicity conjecture. Part II %A Birge Huisgen-Zimmermann %A Manuel Saor¨ªn %J Mathematics %D 2014 %I arXiv %X We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in \Bbb N} M_n$ of finitely generated indecomposable modules $M_n$ is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the $M_n$. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type. %U http://arxiv.org/abs/1407.2365v1