  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 these decompositions: the
\emph{lattice decompositions}. In a first \textit{\'etage} this can be done
using endomorphisms of $M$, which produce a decomposition of the ring
$\textrm{End}_R(M)$ as a product of rings, i.e., they are central idempotent
endomorphisms. But since not every central idempotent endomorphism produces a
lattice decomposition, the classical theory is not of application. In a second
step we characterize when a particular module $M$ has a lattice decomposition;
this can be done, in the commutative case in a simple way using the support,
$\textrm{Supp}(M)$, of $M$; but, in general, it is not so easy. Once we know
when a module decomposes, we look for characterizing its decompositions. We
show that a good framework for this study, and its generalizations, could be
provided by the category $\sigma[M]$, the smallest Grothendieck subcategory of
$\textbf{Mod}-{R}$ containing $M$.
