Weighted Erdős-Burgess and Davenport constant in commutative rings

Let $R$ be a finite commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2=e$. Let $Ψ$ be a subgroup of the group ${\rm Aut}(R)$ of all automorphisms of $R$. The $Ψ-$weighted Erdős-Burgess constant ${\rm I}_Ψ(R)$ 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 $\prod\limits_{i=1}^r ψ_i(a_i)$ is one idempotent of $R$ where $ψ_1,\ldots,ψ_r\in Ψ$. In this paper, for the finite quotient ring of a Dedekind domain $R$, a connection is established between the $Ψ-$weighted-Erdős-Burgess constant of $R$ and the $Ψ-$weighted Davenport constant of its group of units by all the prime ideals of $R$.

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