MAPPINGS BETWEEN LATTICES OF RADICAL SUBMODULES

Let R be a ring and R(M) be the lattice of radical submodules of an R-module M. Although the mapping rho : R(R) -> R(M) defined by rho(I) = rad(IM) is a lattice homomorphism, the mapping sigma : R(M) -> R(R) defined by sigma(N) = (N : M) is not necessarily so. In this paper, we examine the properties of sigma, 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 sigma is a homomorphism. In particular, a finitely generated module M over a domain R is a faithful multiplication module if and only if sigma is an isomorphism.