We examine the properties of certain mappings between the lattice
of ideals of a commutative ring R and the lattice of submodules of an
R-module M, in particular considering when these mappings are lattice homomorphisms.
We prove that the mapping λ from the lattice of ideals of R
to the lattice of submodules of M defined by λ(B) = BM for every ideal B
of R is a (lattice) isomorphism if and only if M is a finitely generated faithful
multiplication module. Moreover, for certain but not all rings R, there is an
isomorphism from the lattice of ideals of R to the lattice of submodules of an
R-module M if and only if the mapping λ is an isomorphism.
Lattice homomorphism, commutative ring, Prufer domain, multiplication module, Noetherian ring