Lattice decomposition of modules

TL;DR

This paper investigates conditions under which the lattice of submodules of a module decomposes into a product, linking module decompositions with endomorphism ring structures and category theory.

Contribution

It characterizes lattice decompositions of modules via endomorphisms and category theory, extending classical results to more general settings.

Findings

Lattice decompositions relate to central idempotent endomorphisms.

Support of modules helps determine decomposability in the commutative case.

Category σ[M] provides a framework for studying module decompositions.

Abstract

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 $M$.