Some results on null ideals of finite rings

Abstract: For a finite associative unital ring $R$, the null ideal of $R$ is the collection of polynomials with coefficients from $R$ that send each element of $R$ to zero under evaluation. It was conjectured that the null ideal of $R$ is always a two-sided ideal of its overlying polynomial ring. The conjecture was proved to be false with the construction of a subring of $4 \times 4$ upper triangular matrices over $\mathbb{F}_2$ for which the null ideal is not two-sided. The Jacobson radical of this counterexample ring has nilpotency 4. We prove that if the Jacobson radical of $R$ has nilpotency at most 3, then the null ideal of $R$ is two-sided. By extending the known counterexample ring, for each $n \geq 5$ we present a ring for which the null ideal is not two-sided, and the Jacobson radical has nilpotency $n$.