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.
Categories: math.CO math.NT