AI 中文总结
该研究针对约化环和主理想整环的商环两类有限交换环，借助新提出的积分覆盖图组合工具，确定了其零因子图的盒维数与阈值维数，解答了相关领域学者近期提出的两个公开问题。
AI 中文摘要
有限交换环$R$的零因子图$\Gamma(R)$以$R$的非零零因子为顶点，当且仅当两个元素的乘积为零时，二者之间存在一条边。我们针对两类有限交换环——约化环与主理想整环的商环——确定了其零因子图$\Gamma(R)$的盒维数与阈值维数。我们的证明使用了一种新的组合工具“积分覆盖图（integral covering graph）”，它捕捉了这两类环共有的结构，并且推广了$[n]$子集上的不交图（disjointness graph，两个子集相邻当且仅当它们不交）。借此，我们解答了L. Sunil Chandran与Suraj Kumar Sahoo近期在《Boxicity of Zero Divisor Graphs》（Discrete Applied Mathematics 391 (2026)）中提出的两个问题。
英文摘要
The zero divisor graph $Γ(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $Γ(R)$ for two classes of finite commutative rings: reduced rings and quotients of principal ideal domains. Our proofs use a new combinatorial gadget, the integral covering graph, that captures the structure shared by both ring families and generalizes the disjointness graph on subsets of $[n]$, where two subsets are adjacent if and only if they are disjoint. In doing so, we answer two questions recently posed by L.~Sunil Chandran and Suraj Kumar Sahoo in Boxicity of Zero Divisor Graphs, Discrete Applied Mathematics 391 (2026).
Comments25 pages 5 figures
