The decomposition of a module M in a direct sum M=M1+M2 does not produce, in general, a decomposition of the lattice L(M) of all submodules in a direct product of lattices L(M)=L(M1)xL(M2). When this happens we say M=M1+M2 is a lattice decomposition. These particular decompositions have special properties: for instance, M1, and M2, are complemented distributive submodules. The main aim of this paper is to characterize lattice decompositions of a module M, over a commutative ring A, in terms of Supp(M). Thus, we show that there is a one-to-one correspondence between lattice decompositions of M and partitions of Supp(M) in two subset closed under specialization satisfying some extra property. In the case where A is a noetherian ring, these partitions are only closed under specializations, but for general rings these extra conditions are necessary for more general rings. On the other hand, if M has a lattice decomposition, then the simple modules which are subfactors of M produce a decomposition of sigma[M], the category of all modules subgenerated by M, in a product of two subcategories.

We prove that lattice decomposition is a local property and also show several applications of the lattice decomposition to the module structure as well as its behaviour in relation to some module constructions, change of ring and ring extensions.