Lower bound for the Erdős-Burgess constant of finite commutative rings

Abstract

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.