Topologically semisimple and topologically perfect topological rings

Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, some equivalences and implications between which we prove. Considering the rings of endomorphisms of modules as topological rings in the finite topology, we establish a close connection between the conjectural concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. In particular, we show that a module $\Sigma$-coperfect over its endomorphism ring has a perfect decomposition provided that the endomorphism ring is commutative, and that all countably indexed local direct summands are direct summands in any countably generated endo-$\Sigma$-coperfect module.