Characterizations of graded Pr\"ufer $\star$-multiplication domains, II

TL;DR

This paper explores the properties of graded Pr"ufer $ ext{star}$-multiplication domains, establishing equivalences with certain localizations and characterizing when these localizations are PIDs.

Contribution

It provides new characterizations of graded Pr"ufer $ ext{star}$-multiplication domains via properties of associated polynomial localizations and extends previous work in the area.

Findings

R is a graded Pr"ufer-$ ext{star}$-multiplication domain iff $ ext{NA}(D, ext{star})$ is a Pr"ufer domain.

R is a graded Pr"ufer-$ ext{star}$-multiplication domain iff $ ext{NA}(R, ext{star})$ is a Be9zout domain.

Conditions under which $ ext{NA}(R,v)$ is a PID are determined.

Abstract

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain and $\star$ be a semistar operation on $R$. For $a\in R$, denote by $C(a)$ the ideal of $R$ generated by homogeneous components of $a$ and for$f=f_0+f_1X+\cdots+f_nX^n\in R[X]$, let $\A_f:=\sum_{i=0}^nC(f_i)$. Let $N(\star):=\{f\in R[X]\mid f\neq0\text{and}\A_f^{\star}=R^{\star}\}$. In this paper we study relationships between ideal theoretic properties of $\NA(R,\star):=R[X]_{N(\star)}$ and the homogeneous ideal theoretic properties of $R$. For example we show that $R$ is a graded Pr\"ufer-$\star$-multiplication domain if and only if $\NA(D,\star)$ is a Pr\"ufer domain if and only if $\NA(R,\star)$ is a B\'ezout domain. We also determine when $\NA(R,v)$ is a PID.