有限环的幂零乘积概率

中文摘要 AI 辅助

对于有限环 $R$，我们研究随机选取 $R$ 中两个元素的乘积为幂零的概率，称之为 $R$ 的幂零乘积概率（NPP），记为 $P_{nil}(R)$。我们推导出依赖于 $R$ 结构的 $P_{nil}(R)$ 的界，并证明在所有含单位元的交换环中，剩余域为 $\mathbb{Z}_{2}$ 的局部环类达到 NPP 的最大值，该值等于 $3/4$。进一步，我们确定含单位元交换环的 NPP 的所有可能取值集合，并分类所有达到这些值的环。最后，我们研究 $R$ 的 NPP 与 $R$ 上矩阵环的某些子环的 NPP 之间的关系。

英文摘要

For a finite ring $R$, we investigate the probability that the product of two randomly chosen elements in $R$ is nilpotent, which we call the nilpotent product probability (NPP) of $R$ and denote by $P_{nil}(R)$. We derive bounds for $P_{nil}(R)$ that depend on the structure of $R$, and prove that among all commutative rings with identity, the class of local rings with residue field $\mathbb{Z}_{2}$ attains the maximum value of NPP, which is equal to $3/4$. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of $R$ and NPP of some subrings of the matrix ring over $R$.

发表机构

• National Institute of Technology Meghalaya（梅加拉亚邦国家技术学院）

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