We examine the properties of certain mappings between the lattice L(R) of ideals of a commutative ring R and the lattice L(RM) of submodules of an R-module M, in particular considering when these mappings are complete homomorphisms of the lattices. We prove that the mapping λ from L(R) to L(RM) defined by λ(B) = BM for every ideal B of R is a complete homomorphism if M is a faithful multiplication module. A ring R is semiperfect (respectively, a finite direct sum of chain rings) if and only if this mapping λ : L(R) → L(RM) is a complete homomorphism for every simple (respectively, cyclic) R-module M. A Noetherian ring R is an Artinian principal ideal ring if and only if, for every R-module M, the mapping λ : L(R) → L(RM) is a complete homomorphism.