MAPPINGS BETWEEN MODULE LATTICES

Abstract

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.

Keywords

• Lattice homomorphism, commutative ring, Prufer domain, multiplication module, Noetherian ring

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11. Department of Mathematics University of Glasgow Glasgow G12 8QW, Scotland UK e-mail: [e-mail]