Are ideal covering numbers always of the form p^d + 1?

Determine whether, for every ring R without identity with finite ideal covering number, each of η_ℓ(R), η_r(R), and η(R) must equal p^d + 1 for some prime p and integer d ≥ 1; if so, classify the possible exponents d, with particular emphasis on the two-sided ideal covering number η(R).

Background

The paper proves that many examples attain the lower bound p+1, derived from the additive group structure R^+, and provides infinite families achieving η_ℓ = p+1, η_r = p+1, and in some cases η = p+1.

This raises the broader question of whether all finite ideal covering numbers must be one more than a prime power, and if so, which prime-power exponents d can occur—especially for the two-sided invariant η(R), where constraints may differ from the one-sided cases.