有限环的零理想的若干结果
Some results on null ideals of finite rings
中文总结 AI 辅助
本文针对有限环的零理想展开研究，证明当环的Jacobson根幂零指数不超过3时其零理想为双侧理想，还构造出幂零指数为任意不小于5的整数的反例环。
中文摘要 AI 辅助
对于有限含幺结合环$R$，$R$的零理想是指系数取自$R$的多项式在赋值下将$R$中每个元素映射为零的全体多项式构成的集合。曾有猜想称，$R$的零理想总是其所属多项式环的双侧理想。该猜想因构造出$\boldsymbol{F}_2$上$4 \times 4$上三角矩阵的一个子环而被证明为假，该子环的零理想并非双侧理想。该反例环的Jacobson根的幂零指数为4。我们证明：若$R$的Jacobson根的幂零指数不超过3，则$R$的零理想是双侧理想。通过扩展已知的反例环，对每个$n \boldsymbol{\geq} 5$，我们给出一个环，其零理想非双侧理想，且Jacobson根的幂零指数为$n$。
英文摘要
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 $n$.
