On topological M-injective modules

Abstract

Let R be a topological ring and M be a topological R-module. A topological R-module U is called a topological M-injective if for every continuous monomorphism f:K→M, where K is an open submodule of M, and for every continuous homomorphism g:K→U, there exists a continuous homomorphism h:M→U such that hf=g. A topological M-injective module is a generalization of a topological injective module defined by Goldman and Sah [14]. An infinite direct sum of injective modules is not necessary injective. In this paper, the properties of topological M-injective modules are investigated. We prove that an infinite direct sum of topological M-injective modules is also topological M-injective if the direct sum is an open submodule of the direct product of topological M-injective modules.