On the Erd\H{o}s-Burgess constant of the multiplicative semigroup of a factor ring of $\mathbb{F}_q[x]$

Abstract

Let $\mathcal{S}$ be a commutative semigroup endowed with a binary associative operation $+$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The {\sl Erd\H{o}s-Burgess constant} of $\mathcal{S}$ is defined as the smallest $\ell\in \mathbb{N}\cup \{\infty\}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence the sum of whose terms is idempotent. Let $q$ be a prime power, and let $\F_q[x]$ be the polynomial ring over the finite field $\F_q$. Let $R=\F_q[x]\diagup K$ be a quotient ring of $\F_q[x]$ modulo any ideal $K$. We gave a sharp lower bound of the Erd\H{o}s-Burgess constant of the multiplicative semigroup of the ring $R$, in particular, we determined the Erd\H{o}s-Burgess constant in the case when $K$ is the power of a prime ideal or a product of pairwise distinct prime ideals in $\F_q[x]$.