Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings

Let $R$ be a finite associative ring with identity, and let $\text{Nul}(R)$ denote the set of polynomials in a central indeterminate that vanish identically on $R$ under right evaluation. We establish two results concerning the failure of $\text{Nul}(R)$ to be a right ideal. First, if $e$ is an idempotent, $f=1-e$, and either $eRf=0$ or $fRe=0$, then right multiplication by $e$ preserves $\text{Nul}(R)$. Thus both off-diagonal Peirce components must be nonzero whenever $e$ witnesses a failure of right stability. Second, if $J$ is a finite nilpotent ideal whose additive group is a $2$-group and $e,f$ are complementary idempotents such that $$eJf\mathrm{\ne }0,fJe\mathrm{\ne }0,J^3\mathrm{\ne }0,$$ then $\mathrm{\mid }J\mathrm{\mid }\ge 32$. Werner’s theorem that $\text{Jac}(R{)}^3=0$ implies the two-sidedness of $\text{Nul}(R)$ therefore yields $\mathrm{\mid }R\mathrm{\mid }\ge 128$ whenever $\text{Nul}(R)$ is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.