Conditions for equality or strict inequality in the covering-number inequality chain

Identify necessary and sufficient conditions for equality or for strict inequality to occur at each step of the chain σ(R^+) ≤ σ(R) ≤ η_ℓ(R), η_r(R) ≤ η(R) for rings R without identity.

Background

Remark 1.1 establishes the comparison chain between the covering number of the additive group σ(R^+), the subring covering number σ(R), and the ideal covering numbers η_ℓ(R), η_r(R), and η(R). The paper’s constructions and computational data show that equalities and strict inequalities can each occur in various positions of the chain.

A complete characterization of when each inequality becomes equality or remains strict would clarify the relationships between additive, subring, and ideal coverings across classes of nonunital rings.