Topologically semisimple and topologically perfect topological rings

Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to realm topological with right linear topology, we show that abelian category left contramodules over such a ring is split (equivalently, semisimple) if and only discrete modules same semisimple). Our results in this direction complement those Iovanov-Mesyan-Reyes. An extension Bass perfect formulated as list conjecturally equivalent conditions, many equivalences implications between which prove. In particular, all conditions are for countable base neighborhoods zero topologically coherent rings. Considering endomorphisms finite establish close connection concept decomposition. also apply endomorphism direct sum decompositions objects certain additive categories more general than modules; call them agreeable categories. We any Grothendieck semisimple. prove module $\Sigma$-coperfect its has decomposition provided either commutative or countably generated, partially answering question Angeleri Hugel Saorin.