Davenport constant for semigroups II

Abstract

Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $\mathrm{D}(\mathcal{S})$, 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 $T^{\prime }$ ($T^{\prime }\ne T$) with the sum of all terms from $T^{\prime }$ 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 $$\mathrm{D}({\mathcal{S}}_R)=\mathrm{D}(U({\mathcal{S}}_R)),$$ where ${\mathcal{S}}_R$ denotes the multiplicative semigroup of the ring $R$, and $U({\mathcal{S}}_R)$ denotes the group of units in ${\mathcal{S}}_R$.

In press in Journal of Number Theory. arXiv admin note: text overlap with arXiv:1409.1313 by other authors

References in corpus (1)

• Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$