Existence of Erdős-Burgess constant in commutative rings

Abstract

Let R be a commutative unitary ring. An idempotent in R is an element e ∈ R with e 2 = e. The Erdős-Burgess constant associated with the ring R is the smallest positive integer ℓ (if exists) such that for any given ℓ elements (not necessarily distinct) of R, say a 1 , . . . , a ℓ ∈ R, there must exist a nonempty subset J ⊂ {1, 2, . . . , ℓ} with j∈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.