Indecomposable Decompositions of Pure-Injective Objects and the Pure-Semisimplicity

We give a criterion for the existence of an indecomposable decomposition of pure-injective objects in a locally finitely presented Grothendieck category (Theorem 2.5). As a consequence we get Theorem 3.2, asserting that an associative unitary ring R is right pure-semisimple if and only if every right R-module is a direct sum of modules that are pure-injective or countably generated. Some open problems are formulated in the paper.