M-injective hulls of topological modules

Let R be a topological ring and M be a topological R-module. An R-module E is called topological M-injective if, for every continuous monomorphism f: L → M with L an open submodule of M, and for every continuous homomorphism g: L → E, there exists a continuous homomorphism h: M → E that extends f. A topological M-injective hull of N is a minimal topological M-injective extension of N. If (N,τ) is a topological module in the category Topσ[M], then N has injective hulls in the categories R-MOD, σ[M], TopR-MOD, and Topσ[M], which are not necessarily the same in general. A topological M-injective hull of N in Topσ[M] is the trace of M in Eτ(N), where Eτ(N) is a topological injective hull of N in TopR-MOD. If E(N), Tr(M,E(N)), Eτ(N), and Tr∗(M,E τ(N)) are injective hulls of N in R-MOD, σ[M], TopR-MOD, and Topσ[M], respectively, then N ⊆Tr∗(M,E τ(N)) ⊆ Eτ(N) ⊆ E(N) and N ⊆Tr∗(M,E τ(N)) ⊆Tr(M,E(N)) ⊆ E(N). Furthermore, we prove that an infinite direct sum of topological M-injective hulls is homeomorphic to a topological M-injective hull of the direct sum of topological R-modules in Topσ[M] if the direct sum of topological M-injective hulls is an open submodule in the direct product of topological M-injective hulls.