On the uniqueness of topologies for some constructions of rings and modules
It is well known that upon passage from rings to their superrings a topology given on a ring, even extendible to a corresponding superring, generally extends in a nonunique fashion. For this reason, it is of interest to indicate the cases in which the question of uniqueness is resolved affirmatively. It is the study of this question that the present article addresses. The results of \S 1 are auxiliary and are used in the proof of the results of \S 3, although Theorem 1.7 proven in \S 1 is of interest in its own right. As a corollary (see Corollary 1.9) it implies Nachbin's uniqueness theorem for a topology on a finite-dimensional vector space. In \S 2 we discuss the question of coincidence of topologies on a ring which agree on an ideal of the ring and on the quotient ring by the ideal (see Theorem 2.1). The results of this section are also used in \S 4 in studying the rings with a unique compact topology. In \S 3 we study topologies on subrings of the matrix ring over a given topological ring (see Theorems 3.1 and 3.2). Moreover, in the article we construct a series of examples demonstrating that the constraints imposed in the hypotheses of the theorems are essential and cannot be omitted.
scientific article; zbMATH DE number 923532