Davenport constant for semigroups II

Abstract: Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted ${\rm D}(\mathcal{S})$, is defined to be the least positive integer $\ell$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $\ell$ contains a proper subsequence $T'$ ($T'\neq T$) with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $q>2$ be a prime power, and let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$. Let $R$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. We prove that $${\rm D}(\mathcal{S}_R)={\rm D}(U(\mathcal{S}_R)),$$ where $\mathcal{S}_R$ denotes the multiplicative semigroup of the ring $R$, and $U(\mathcal{S}_R)$ denotes the group of units in $\mathcal{S}_R$.