有限交换环的Erdens-Burgess常数的下界
Lower bound for the Erdős-Burgess constant of finite commutative rings
摘要：
设$R$是有限交换酉环. $R$中的幂等元是具有$e^2=e$的元素$e\in R$。与环$R$相关联的Erdens-Burgess常数是最小的正整数$\ell$，使得对于$R$的任何给定的$\ell$元素（允许重复），比如$a_1,\ldots,a_{\ell}\in R$，必须存在非空子集$J\subset \{1,2,\ldots,\ell\}$，其中$\prod\limits_{j\in J} a_j$是幂等元。本文给出了有限交换酉环中Erdens-Burgess常数关于其所有极大理想的一个下界，并证明了在某些情况下可达到这个下界.这个结果统一了最近得到的关于这个不变量的一些定理。
英文摘要：
Let $R$ be a finite commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2=e$. The Erdős-Burgess constant associated with the ring $R$ is the smallest positive integer $\ell$ such that for any given $\ell$ elements (repetitions are allowed) of $R$, say $a_1,\ldots,a_{\ell}\in R$, there must exist a nonempty subset $J\subset \{1,2,\ldots,\ell\}$ with $\prod\limits_{j\in J} a_j$ being an idempotent. In this paper, we give a lower bound of the Erdős-Burgess constant in a finite commutative unitary ring in terms of all its maximal ideals, and prove that the lower bound is attained in some cases. The result unifies some recently obtained theorems on this invariant.
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