Let R be a ring and R(M) be the lattice of radical submodules of an R-module M. Although the mapping ρ : R(R) → R(M) defined by ρ(I) = rad(IM) is a lattice homomorphism, the mapping σ : R(M) → R(R) defined by σ(N) = (N : M) is not necessarily so. In this paper, we examine the properties of σ, in particular considering when it is a homomorphism. We prove that a finitely generated R-module M is a multiplication module if and only if σ is a homomorphism. In particular, a finitely generated module M over a domain R is a faithful multiplication module if and only if σ is an isomorphism.