Extending Ring Topologies

Abstract

Throughout this abstract, (R,T) denotes a compact (Hausdorff) topological ring. The authors extend T to ring topologies on R which are totally bounded or even pseudocompact; a principal tool is the Bohr compactification of a topological ring. They show inter alia: If the Jacobson radical J(R) of R satisfies w(R/J(R))>ω then there is a pseudocompact ring topology on R strictly finer than T; if in addition w(R)=w(R/J(R))=α with cf(α)>ω then there are exactly 22|R|-many such topologies.The ring (R,T) is said to be a van der Waerden ring if T is the only totally bounded ring topology on R. Theorem 4.13 asserts that if R is semisimple, then R is a van der Waerden ring if and only if in the Kaplansky representation R=Πn<ω(Rn)αn of R as a product of full matrix rings over finite fields each αn is finite.Other classes of van der Waerden rings are constructed, and it is shown that there are non-compact totally bounded rings (S,U) such that U is the only totally bounded ring topology on S.