Determine covering numbers of modules

Determine the set of integers that can occur as covering numbers of modules, where for a ring R and an R‑module M the covering number is defined as the minimal cardinality of a collection of proper R‑submodules whose union equals M. Concretely, classify all possible covering numbers for modules over arbitrary (commutative or noncommutative) rings, extending beyond the known case for modules over commutative unital rings (where the covering number is q+1 for some prime power q) and the noncommutative examples constructed for modules over M_n(F_q).

Background

The paper classifies rings that can be covered by finitely many proper left, right, or two-sided ideals and determines the corresponding ideal covering numbers. As a byproduct, the authors construct modules over noncommutative rings M_n(F_q) whose covering numbers by submodules equal (q^{n+1}−1)/(q−1), providing the first such examples beyond the commutative setting.

Prior work in the commutative case (Khare–Tikaradze, Thm. 2.2) shows that if a module over a commutative unital ring admits a finite cover by proper submodules, then its covering number is q+1 for some prime power q. The present results indicate that the landscape for modules—especially over noncommutative rings—is broader, and a full classification of possible covering numbers remains unresolved.