We study the following question: for which Abelian groups do their rings of endomorphisms admit a compact ring topology? Fuchs [4] asked this question for Abelian p-groups. A classification of torsion Abelian groups A of cardinality <2ω and such that &#x1d508;&#x1d52b;&#x1d521;(A) admits a compact ring topology is given. We construct a torsion complete p-group of cardinality 2ω whose ring of endomorphisms admits a compact ring topology. Some facts about non-torsion Abelian groups whose rings of endomorphisms admit a compact ring topology are derived. We also construct an Abelian group A for which the Jacobson radical J(&#x1d508;&#x1d52b;&#x1d521;(A)) is not closed with respect to the finite topology. Motivated by this example, we introduce a new radical for topological rings and derive some of its basic properties.

Topological Rings of Endomorphisms

Semigroup rings with certain finiteness conditions

The structure of the semigroup S is investigated when the semigroup ring RS admits a compact topology. It is proved that, in case of the semisimple semigroup S , the ”last” principal factors of S have finitely many of right or left ideals. It is shown that it is not true for other factors.

I.B. Kozhukhov

Papers

On the compact semigroup rings