Generalized Morita equivalence for linearly topologized rings
Two topological rings (A,\(\sigma)\) and (R,\(\tau)\) having bases at zero consisting of right ideals \((=\) linearly topologized) are called similar if the categories \(Mod\)-A\({}_{\sigma}\) and \(Mod\)-R\({}_{\tau}\) of discrete (A,\(\sigma)\)- and (R,\(\tau)\)-modules are equivalent. The author observes that every linearly topologized ring is similar to its Hausdorff completion. For some notions which are well known in the case of discrete rings their topological modifications are given [for example, for quasi- progenerator (cf. \textit{K. R. Fuller}, J. Algebra 29, 528-550 (1974; Zbl 0306.16020)), functors Hom and tensor product]. Using these modifications a characterization of similarities is given for complete linearly topologized rings, which is analogous to the description obtained for discrete rings. Moreover, it is proved that if two commutative complete linearly topologized rings are similar then there exists an (algebraic) isomorphism \(\alpha\) : \(A\to B\) such that an ideal \(I\subset A\) is closed if and only if \(\alpha\) (I) is closed in B.
scientific article; zbMATH DE number 4079683