Existence of Erdős-Burgess constant in commutative rings

Abstract

Let $R$ be a 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 $\ell$ (if exists) such that for any given $\ell$ elements (not necessarily distinct) of $R$, say $a_1,\dots ,a_{\ell }\in R$, there must exist a nonempty subset $J\subset \{1,2,\dots ,\ell \}$ with $\prod _{j\in J}{a}_j$ being an idempotent. In this paper, we prove that except for an infinite commutative ring with a very special form, the Erdős-Burgess constant of the ring $R$ exists if and only if $R$ is finite.

8 pages. arXiv admin note: text overlap with arXiv:2002.11489