On graded Going down domains

TL;DR

This paper introduces the concept of graded going-down domains in the context of $ ext{Γ}$-graded integral domains, providing characterizations, equivalent conditions, and stability properties relevant to graded algebraic structures.

Contribution

It defines graded going-down domains, establishes their relation to graded Prüfer and finite-conductor domains, and proves stability under factor domains.

Findings

Characterization of graded going-down domains via graded divided domains.

Equivalent conditions for graded Prüfer domains involving graded going-down.

Stability of the graded going-down property under factor domains.

Abstract

Let $\Gamma$ be a torsionless commutative cancellative monoid and $R =\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain. In this paper, we introduce the notion of graded going-down domains. Among other things, we provide an equivalent condition for graded-Pr\"{u}fer domains in terms of graded going-down and graded finite-conductor domains. We also characterize graded going-down domains by means of graded divided domains. As an application, we show that the graded going-down property is stable under factor domains.