The direct sum decomposition of a module M in a direct sum M= M 1⊕ M 2 does not
produce, in general, a decomposition of the lattice ℒ (M) of all submodules of M in a direct …

The first aim of this work is to characterize when the lattice of all submodules of a module is
a direct product of two lattices. In particular, which decompositions of a module M produce …