Two-sided ideal covering number for a specific 3×3 matrix ring over F_q

Determine whether the two-sided ideal covering number η(R)—the minimal number of proper two-sided ideals whose union equals R—equals q+1 for a specific subring R of M_3(F_q) (3×3 matrices over the finite field F_q with q = p^d elements) that is known to satisfy η_ℓ(R) = q+1 and for which a closely related ring R′ satisfies η_r(R′) = q+1.

Background

The paper introduces three ideal covering numbers for a ring R (not assumed unital or commutative): η_ℓ(R), η_r(R), and η(R), the minimal numbers of proper left, right, and two-sided ideals whose unions equal R, respectively.

In Section 3, the authors report a construction, suggested by Nicholas Werner, of a ring R formed by 3×3 matrices over F_q for which computational evidence indicates η_ℓ(R) = q+1; a related ring R′ has η_r(R′) = q+1. What remains unresolved is whether the two-sided invariant η(R) matches this prime-power-plus-one value for that same R.

Resolving this would clarify whether such matrix-based constructions can simultaneously achieve the sharp lower bound for both one-sided and two-sided ideal coverings in this family.