Existence of covering topological R-modules

Abstract

Let R be a topological ring with identity and M a topological (left) R-module such that the underlying topology of M is path connected and has a universal cover. Let 0 ∈ M be the identity element of the additive group structure of M, and N a submodule of the R-module π1(M,0). In this paper we prove that if R is discrete, then there exists a covering morphism p: (e MN, ˜ 0) → (M,0) of topological R-modules with characteristic group N and such that the structure of R-module on M lifts to e MN. In particular, if N is a singleton group, then this cover becomes a universal cover.