Extending Ring Topologies

Throughout this abstract, (R, ) denotes a compact (Hausdorff) topological ring. The authors extend 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 ; if in addition w(R) = w(R/J(R)) = α with cf(α) > ω then there are exactly 22R-many such topologies. The ring (R, ) is said to be a van der Waerden ring if 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, ) such that is the only totally bounded ring topology on S.