Existence of Erd\H{o}s-Burgess constant in commutative rings

TL;DR

This paper investigates the Erd ext{"o}s-Burgess constant in commutative rings, establishing that it exists only in finite rings except for a specific class of infinite rings.

Contribution

It proves a characterization of when the Erd ext{"o}s-Burgess constant exists in commutative rings, linking existence to the finiteness of the ring.

Findings

Erd ext{"o}s-Burgess constant exists iff the ring is finite, with exceptions.

Infinite rings with a special form do not have this constant.

The result provides a complete criterion for the existence of the 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 Erd\H{o}s-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 $a_1,\ldots,a_{\ell}\in R$, 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 prove that except for an infinite commutative ring with a very special form, the Erd\H{o}s-Burgess constant of the ring $R$ exists if and only if $R$ is finite.