Representation of Homomorphisms Between Submodule Lattices

Abstract

Under certain weak conditions on the module RM every mapping f: L(RM) → L(SN) between the submodule lattices, which preserves arbitrary sums (= joins) and ‘disjointness’, has a unique representation of the form f (U) = 〈h[ SBR × RU]〉for all U ∈ L(RM), where SBR is some bimodule and h is an R-balanced mapping. Furthermore f is a lattice homomorphism if and only if BR is flat and the induced S-module homomorphism h:SB⊗RM → SN is monic. If SN satisfies also the same weak conditions then f is a lattice isomorphism if and only if BR is a finitely generated projective generator, S - End(BR ) canonically and h: SB⊗ → SN is an S-module isomorphism, i.e. every lattice isomorphism is induced by a Morita equivalence between R and S and a module isomorphism.