Topologically semisimple topological rings

The classical Wedderburn-Artin theorem describes associative rings A for which the category of A-modules is semisimple. This turns out to be a left-right symmetric property: the category of left A-modules is semisimple if and only if the category of right A-modules is. In this talk, I will present a generalization of the Wedderburn-Artin theory to topololgical associative rings R in which open right ideals form a base of neighborhoods of zero. The talk will start with a discussion of split and semisimple abelian categories and end with a description of topological rings R for which the category of left R-contramodules is split (or equivalently, semisimple) or, equivalently, the category of discrete right R-modules is split (or equivalently, semisimple).