**Leonid Positselski, Czech Academy of
Science, Prague**

**Title**: **Countably generated endo-Sigma-coperfect modules
have perfect decomposition**

**Abstract**: In a 2006 paper of Angeleri Hugel and Saorin, it
was shown that any module with a perfect decomposition is
Sigma-coperfect over its endomorphism ring, and a question was asked
whether the converse implication holds. In this talk, I will present
a topological algebra approach to this problem. The ring of
endomorphisms of any module is endowed with the so-called finite
topology, making it a complete, separated topological ring with a
base of neighborhoods of zero formed by open right ideals. Both the
existence of a perfect decomposition and the
endo-Sigma-coperfectness properties of a module are shown to be
equivalent to certain properties of the topological ring of
endomorphisms. The equivalence of these two properties of a
topological ring is an assertion extending Bass' famous Theorem P to
the realm of topological rings. One can prove it for commutative
topological rings and for topological rings with a countable base of
neighborhoods of zero. It follows that the answer to the question of
Angeleri Hugel and Saorin is positive for modules with a commutative
endomorphism ring and for countably generated modules. This talk is
based on a joint work of Jan Stovicek and the speaker.