On Zimmermann-Huisgen’s splitting theorem

Abstract:

This note is motivated by a paper of Birge Zimmermann-Huisgen, which in turn is motivated by a long sequence of papers—the first due to Faith—dealing with the question of when the canonical embedding of a direct sum of modules in the corresponding direct product splits. Zimmermann-Huisgen answered a question raised by previous authors by showing that if $R$ is a von Neumann regular ring the only way this can happen is that, except for a finite number, the modules involved must each be semisimple with only a finite number of simple modules involved. Based on a new, more elementary argument, we establish a necessary condition for the sum-product splitting over an arbitrary (associative) ring ft (with identity).