# Covering modules by proper submodules

Apoorva Khare and Akaki Tikaradze

Arxiv ID: 0906.1023•Last updated: 2/9/2022

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.

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