  We define topologically semiperfect (complete, separated, right linear)
topological rings and characterize them by equivalent conditions. We show that
the endomorphism ring of a module, endowed with the finite topology, is
topologically semiperfect if and only if the module is decomposable as an
(infinite) direct sum of modules with local endomorphism rings. Then we study
structural properties of topologically semiperfect topological rings and prove
that their topological Jacobson radicals are strongly closed and the related
topological quotient rings are topologically semisimple. For the endomorphism
ring of a direct sum of modules with local endomorphism rings, the topological
Jacobson radical is described explicitly as the set of all matrices of
nonisomorphisms. Furthermore, we prove that, over a topologically semiperfect
topological ring, all finitely generated discrete modules have projective
covers in the category of modules, while all lattice-finite contramodules have
projective covers in both the categories of modules and contramodules. We also
show that the topological Jacobson radical of a topologically semiperfect
topological ring is equal to the closure of the abstract Jacobson radical, and
present a counterexample demonstrating that the topological Jacobson radical
can be strictly larger than the abstract one. Finally, we discuss the problem
of lifting idempotents modulo the topological Jacobson radical and the
structure of projective contramodules for topologically semiperfect topological
rings.
