We show that under certain weak conditions on the module _RM, every mapping f : ￡(_RM) → ￡(_SN) between the submodule lattices which preserves arbitrary joins and "disjointness" has a unique representation of the form f(u) = for all u ∈￡(_RM), where _SB_R 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 also satisfies the same weak conditions, then f is a lattice isomorphism if and only if Br is a finitely generated projective generator, S ≌ End(B_R) canonically, and h : _SB_⊕ _RM → _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.
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