On Maximal Chains in the Lattice of Module Topologies

Abstract

Let (R,τ R ) be a topological ring and R M, a left unitary R-module. The set L(M) of all (R,τ R )-module topologies on R M is a complete modular lattice. A topology τ∈L(M) is n-premaximal if in L(M) there exists an inclusion-maximal chain τ0>τ1>...> τ n such that τ0 is the largest element in L(M) and τ n =τ. Section 1 contains conditions for existence of 1-premaximal Hausdorff topologies on R M; Section 2 contains a description of all n-premaximal topologies in the case when (R,τ R ) is a topological skew field whose topology is determined by a real absolute value.