Topological Rings of Bounded and Compact Group Homomorphisms on a Topological Ring

Abstract

Let X be a topological ring. In this paper, we consider the three classes of nr-bounded, br-bounded and cr-continuous group homomorphisms defined on X and denote these classes by Bnr(X);Bbr(X) and Bcr(X), respectively. We show that the operations of addition and product in Bnr(X);Bbr(X) and Bcr(X) are continuous with respect to the topology considered on them. Moreover, we introduce the classes of nr-compact and br-compact group homomorphisms, and we study the situations where a class of bounded group homomorphisms coincides with the corresponding class of compact group homomorphisms.