On topological M-injective modules

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. (c) 2023 Elsevier B.V. All rights reserved.