Algebraic representation of mappings between submodule lattices

We show that under certain weak conditions on the module _R M, every mapping between the submodule lattices which preserves arbitrary joins and “disjointness” has a unique representation of the form f(u) = 〈h _S B _R × _R U]〉 for all u ∈ , where _S B _R is some bimodule and h is an R-balanced mapping. Furthermore, f is a lattice homomorphism if and only if B _R is flat and the induced S-module homomorphism is monic. If _S N also satisfies the same weak conditions, then f is a lattice isomorphism if and only if B _R is a finitely generated projective generator, S ≅ End(B _R ) canonically, and is an S-module isomorphism, i.e., every lattice isomorphism is induced by a Morita equivalence between R and S and a module isomorphism. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 46, Algebra, 2007.