Topological Rings of Endomorphisms

Abstract

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 𝔈𝔫𝔡(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(𝔈𝔫𝔡(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.