Extending ring topologies
The authors study the question of strengthening totally bounded (= precompact and Hausdorff) ring topologies on rings. The main tool is the concept of the Bohr compactification of a topological ring. It is established for a given discrete ring \(R\) an antiisomorphism between the lattice of precompact topologies on \(R\) and the lattice of closed ideals of its Bohr compactification \(bR\). There are given suffient conditions for the existence for a compact ring \((R,{\mathfrak T})\) of pseudocompact ring topologies extending \(\mathfrak T\): Theorem. Let \((R,{\mathfrak T})\) be a compact ring for which \(\alpha:=w(R,{\mathfrak T})\) satisfies \(\text{cf}(\alpha)>\omega\). Assume that either (i) \(w(R/J(R))=\alpha\), or (ii) \(R\) is semisimple. Then there exist exactly \(2^{2^{|R|}}\)-many pseudocompact topologies \(\mathfrak V\) on \(R\) such that \({\mathfrak V}\supseteq{\mathfrak T}\). A topological ring \((R,{\mathfrak T})\) is said to be a van der Waerden ring provided each ring homomorphism \(\alpha\) in any compact ring \((R',{\mathfrak T}')\) is continuous. It is given in the paper a complete characterization of semisimple in the sense of Jacobson compact van der Waerden rings. It is proved that compact Noetherian rings are van der Waerden rings. It is also proved that a matrix ring over a van der Waerden ring with identity is a van der Waerden ring. In the last part of the paper is calculated the Bohr compactification of \(\mathbb{Z}\) and are given different examples of van der Waerden rings. The authors make interesting remarks concerning the analogy between the results of the paper under review and the results concerning Abelian groups contained in earlier papers. They note (page 33), for example, that in contrast with the case of Abelian groups, there exist van der Waerden rings having non-closed ideals.
scientific article; zbMATH DE number 1539373
scientific article