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 2 2 R -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 = n<ω R n α n of R as a product of full matrix rings over finite fields each α n 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.

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