Covering modules by proper submodules

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.