Pseudocompact refinements of compact ring topologies

Abstract

In this paper, results of Tsarelunga resp. Comfort, Szambien and the first-listed author are improved. Throughout this abstract, (R,T) denotes a nonmetrizable compact ring. First a main tool is shown: If (R,T) is topologically nilpotent, then w(R)=w(R/R2‾) holds. By using tensor products of unitary modules it is proved that every nonmetrizable compact ring with an identity has a proper pseudocompact refinement. (R,T) admits exactly 22|R|-many pseudocompact ring topologies on R finer than T in the following cases: R is a commutative local ring; (R,T) is topologically nilpotent; (R,T) is commutative such that w(R,T) is a regular cardinal number.