Davenport constant of the multiplicative semigroup of the ring ${\mathbb{Z}}_{n_1}\oplus \cdots \oplus {\mathbb{Z}}_{n_r}$

Abstract

Given a finite commutative semigroup $\mathcal{S}$ (written additively), denoted by $\mathrm{D}(\mathcal{S})$ the Davenport constant of $\mathcal{S}$, namely the least positive integer $\ell$ such that for any $\ell$ elements $s_1,\dots ,s_{\ell }\in \mathcal{S}$ there exists a set $I⊊[1,\ell ]$ for which ${\sum }_{i\in I}{s}_i={\sum }_{i=1}^{\ell }{s}_i$. Then, for any integers $r\ge 1,n_1,\dots ,n_r>1$, let $R={\mathbb{Z}}_{n_1}\oplus \cdots \oplus {\mathbb{Z}}_{n_r}$ be the direct sum of these $r$ residue class rings ${\mathbb{Z}}_{n_1},\dots ,{\mathbb{Z}}_{n_r}$. Moreover, let ${\mathcal{S}}_R$ be the multiplicative semigroup of the ring $R$, and $\mathrm{U}({\mathcal{S}}_R)$ the group of units of ${\mathcal{S}}_R$. In this paper, we prove that $$\mathrm{D}(\mathrm{U}({\mathcal{S}}_R))+P_2\le \mathrm{D}({\mathcal{S}}_R)\le \mathrm{D}(\mathrm{U}({\mathcal{S}}_R))+\delta ,$$ where $P_2=♯\{i\in [1,r]:2\parallel n_i\}$ and $\delta =♯\{i\in [1,r]:2\mid n_i\}.$ This corrects our previous published wrong result on this problem.

10 pages

References in corpus (3)

• A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of $\F_2[x]$
• Structure of the largest idempotent-product free sequences in semigroups
• Additively irreducible sequences in commutative semigroups

Cited by in corpus (1)

• A stronger connection between the Erdős-Burgess and Davenport constants