Conditions for subring coverability of nonunital rings

Characterize all rings R without a multiplicative identity that are subring coverable, meaning that R can be expressed as a finite union of proper subrings.

Background

Beyond ideals, the paper also considers subring coverings, denoted σ(R), and relates them to ideal coverings via the chain σ(R^+) ≤ σ(R) ≤ η_ℓ(R), η_r(R) ≤ η(R).

While coverability by proper ideals necessarily excludes rings with identity, subring coverability is more permissive and has been partly analyzed in the literature for small covering numbers (e.g., unions of three subrings). A general structural characterization for nonunital rings is not provided in the paper, motivating this open problem.