A right pure semisimple ring is an associative ring with identity for which every unital right module is a direct sum of finitely generated modules. Equivalently, every right module is a direct sum of finitely generated indecomposable modules. Left pure semisimplicity is defined with left modules instead (Šaroch 2021).
The summands need not be simple or projective. In contrast, over a semisimple ring, every module is a direct sum of simple modules. Every semisimple ring is pure semisimple on both sides. The pure semisimplicity conjecture asks whether right pure semisimplicity always forces the corresponding left-sided property and finite representation type.
