Weighted Erd\H{o}s-Burgess and Davenport constant in commutative rings

Abstract

Let $R$ be a finite commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2=e$. Let $\Psi$ be a subgroup of the group ${\rm Aut}(R)$ of all automorphisms of $R$. The $\Psi-$weighted Erd\H{o}s-Burgess constant ${\rm I}_{\Psi}(R)$ is defined as the smallest positive integer $\ell$ such that every sequence over $R$ of length at least $\ell$ must contain a nonempty subsequence $a_1,\ldots, a_{r}$ such that $\prod\limits_{i=1}^r \psi_i(a_i)$ is one idempotent of $R$ where $\psi_1,\ldots,\psi_r\in \Psi$. In this paper, for the finite quotient ring of a Dedekind domain $R$, a connection is established between the $\Psi-$weighted-Erd\H{o}s-Burgess constant of $R$ and the $\Psi-$weighted Davenport constant of its group of units by all the prime ideals of $R$.