Weighted Erdős-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 $\mathrm{Aut}(R)$ of all automorphisms of $R$. The $\Psi -$weighted Erdős-Burgess constant ${\mathrm{I}}_{\Psi }(R)$ is defined as the smallest positive integer $\ell$ such that every sequence over $R$ of length at least $\ell$ must contain a nonempty subsequence $a_1,\dots ,a_r$ such that $\prod _{i=1}^r{\psi }_i(a_i)$ is one idempotent of $R$ where ${\psi }_1,\dots ,{\psi }_r\in \Psi$. In this paper, for the finite quotient ring of a Dedekind domain $R$, a connection is established between the $\Psi -$weighted-Erdős-Burgess constant of $R$ and the $\Psi -$weighted Davenport constant of its group of units by all the prime ideals of $R$.

arXiv comment18 pages