Covering modules by proper submodules

TL;DR

This paper generalizes the problem of covering groups and vector spaces by proper substructures to modules over rings, providing complete characterizations and calculations for various classes of modules.

Contribution

It introduces a unified framework for covering modules by proper submodules, extending classical results and computing covering numbers for specific module classes.

Findings

Characterizes when the covering number is finite, paralleling Neumann's 1954 result.

Calculates covering numbers for finitely generated modules over quasi-local rings and PIDs.

Determines the covering number for direct sums of cyclic monoids.

Abstract

A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover $R$-modules by proper submodules for commutative rings $R$, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number $\aleph$, possibly infinite, such that a given $R$-module is a union of $\aleph$-many proper submodules. (1) We completely characterize when $\aleph$ is a finite cardinal; this parallels for modules a 1954 result of Neumann. (2) We also compute the covering (cardinal) numbers of finitely generated modules over quasi-local rings and PIDs, recovering past results for vector spaces and abelian groups respectively. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained.