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

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 $\mathrm{\ell }$ (if exists) such that for any given $\mathrm{\ell }$ elements (not necessarily distinct) of $R$, say $a_1,\ldots,a_{\ell}\in R$,theremustexistanonemptysubset$J\subset {1,2,\ldots,\ell}$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.