COMPLETE HOMOMORPHISMS BETWEEN THE LATTICES OF RADICAL SUBMODULES

Let R be a commutative ring and R(M) be the lattice of radical submodules of an R-module M. In this paper, we examine the properties of the mapping sigma : R(M) -> R(R) defined by sigma(N) = (N : M) and the mapping rho : R(R) -> R(M) defined by rho(I) = rad(IM), in particular considering when these are complete homomorphisms of the lattices. It is shown that a finitely generated module M is a multiplication module if and only if sigma is a lattice homomorphism if and only if sigma is a complete lattice homomorphism. It is also proved that for modules over an Artinian ring, finitely generated faithful multiplication modules and projective modules, rho is a complete lattice homomorphism.