Do rings with ideal covering number p+1 always have a factor ring isomorphic to one of the constructed families?

Ascertain whether every ring R for which an ideal covering number equals p+1 (for some prime p) necessarily has a factor ring isomorphic to one of the explicit example families R1, R2, R3, or R4 constructed in this work.

Background

Theorem 1.1 constructs four infinite families of rings (R1–R4) that achieve the sharp lower bound p+1 in various combinations of left, right, and two-sided ideal coverings. Corollary 1.2 fully characterizes the case of ideal covering number 3 via specific factor rings in small matrix rings over Z_2 and Z_4.

This suggests a potential structural principle: perhaps any ring attaining the minimal value p+1 must admit a factor ring isomorphic to one of these canonical examples. The paper leaves open whether such a reduction via factor rings always exists.