  Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant
of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive
integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length
at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$
equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial ring in
one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this
paper, we made a study of the Davenport constant of the multiplicative
semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$. Among
other results, we mainly prove that, for any prime $p>2$ and any polynomial
$f(x)\in \F_p[x]$ which can be factorized into several pairwise non-associted
irreducible polynomials in $\F_p[x]$, then
$$D(\mathcal{S}_{f(x)}^p)=D(U(\mathcal{S}_{f(x)}^p)),$$ where
$\mathcal{S}_{f(x)}^p$ denotes the multiplicative semigroup of the quotient
ring $\frac{\F_p[x]}{\langle f(x)\rangle}$ and $U(\mathcal{S}_{f(x)}^p)$
denotes the group of units of the semigroup $\mathcal{S}_{f(x)}^p$.

Number Theory;Combinatorics