Erdős–Burgess constant in commutative rings

Abstract

Let $\mathcal {S}$ be a nonempty semigroup endowed with a binary associative operation $*$. An element e of $\mathcal {S}$ is said to be idempotent if $e*e=e$. Originated by one question of P. Erdős to D.A. Burgess: If $\mathcal {S}$ is a finite semigroup of order n, does any $\mathcal {S}$-valued sequence T of length n contain a nonempty subsequence the product of whose terms, in some order, is idempotent?, we make a study of the associated invariant, denoted $\mathrm{I}(\mathcal {S})$ and called Erdős–Burgess constant which is the smallest positive integer $\ell$ such that any $\mathcal {S}$-valued sequence T of length $\ell$ contains a nonempty subsequence the product of whose terms, in some order, is an idempotent. Let $\mathcal {S}_R$ be the multiplicative semigroup of any commutative unitary ring R. We prove that $\mathrm{I}(\mathcal {S}_R)$ is finite if and only if R is finite, provided that the quotient ring R/J(R) of R modulo its Jacobson radical J(R) is not a direct product of an infinite Boolean unitary ring and finitely many finite fields. As a consequence, if R is Noetherian, then $\mathrm{I}(\mathcal {S}_R)$ is finite if and only if R is finite.