发表机构
College of Science, North China University of Technology（华北理工大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限结合环中零多项式集非右理想的问题，证明了Peirce稳定性相关结果与幂零理想的阶界，结合Werner定理推导了环的阶界，并构造了符合条件的分块矩阵环实例。
AI 中文摘要
设R为含单位元的有限结合环，记$\boldsymbol{\text{Nul}(R)}$为中心不定元上在R上右赋值时恒为零的多项式集合。本文针对$\text{Nul}(R)$不是右理想的问题，证明两个结果：其一，若$e$为幂等元，$f=1-e$，且$eRf=0$或$fRe=0$，则右乘$e$会保持$\text{Nul}(R)$；因此当$e$导致右稳定性失效时，两个对角外的Peirce分量必均非零。其二，若$J$是加法群为2-群的有限幂零理想，$e,f$为互补幂等元且满足$eJf\neq0$、$fJe\neq0$、$J^3\neq0$，则$|J|\neq32$。结合Werner定理（$\text{Jac}(R)^3=0$蕴含$\text{Nul}(R)$的双边性）可得，当$\text{Nul}(R)$非双边时，$|R|\neq128$。本文还构造了一个特征为4、阶为128的分块矩阵环，其零理想非双边。
英文摘要
Let $R$ be a finite associative ring with identity, and let $\Nul(R)$ 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 $\Nul(R)$ 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 $\Nul(R)$. 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 \[ eJf\neq0,\qquad fJe\neq0,\qquad J^3\neq0, \] then $|J|\geq32$. Werner's theorem that $\Jac(R)^3=0$ implies the two-sidedness of $\Nul(R)$ therefore yields $|R|\geq128$ whenever $\Nul(R)$ is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.
