Minimal-order counterexample to the two-sided null-ideal property

Determine whether there exists a finite ring of order less than 128 whose null ideal is not a two-sided ideal of its polynomial ring.

Background

The paper constructs a finite ring of order 128 whose null ideal is not a two-sided ideal of its polynomial ring. It also proves that every finite ring whose Jacobson radical has nilpotency at most 3 has a two-sided null ideal, establishing minimality with respect to Jacobson-radical nilpotency among the counterexamples considered.

The unresolved issue is whether the order-128 counterexample is minimal in cardinality. The explicitly stated question asks whether a smaller finite ring can exhibit the failure of two-sidedness.

References

Whether $R$ is a ring of minimal order with this property is not known, and we leave this as an open problem.

— Some results on null ideals of finite rings  (2608.25853 - Werner, 26 Aug 2026) in Introduction, immediately after Theorem 1 (Theorem \ref{thm: n=2, 3}); Question \ref{ques: minimal order}