Existence of ErdH{o}s-Burgess constant in commutative rings

Let $R$ be a commutative unitary ring. An idempotent in $R$ is an element $ein R$ with $e^2=e$. The ErdH{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 $Jsubset {1,2,ldots,ell}$ with $prodlimits_{jin J} a_j$ being an idempotent. In this paper, we prove that except for an infinite commutative ring with a very special form, the ErdH{o}s-Burgess constant of the ring $R$ exists if and only if $R$ is finite.