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

TL;DR

This paper extends the theory of Pr"ufer $ ext{star}$-multiplication domains to graded integral domains, introducing graded $ ext{star}$-Nagata and Kronecker rings and providing new characterizations, including for P$v$MDs.

Contribution

It defines graded $ ext{star}$-Nagata and Kronecker rings and offers new equivalent conditions for graded Pr"ufer $ ext{star}$-multiplication domains, including characterizations for P$v$MDs.

Findings

Established graded $ ext{star}$-Nagata and Kronecker rings.

Provided new characterizations for graded Pr"ufer $ ext{star}$-multiplication domains.

Extended classical ungraded theory to graded integral domains.

Abstract

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain graded by an arbitrary grading torsionless monoid $\Gamma$, and $\star$ be a semistar operation on $R$. In this paper we define and study the graded integral domain analogue of $\star$-Nagata and Kronecker function rings of $R$ with respect to $\star$. We say that $R$ is a graded Pr\"{u}fer $\star$-multiplication domain if each nonzero finitely generated homogeneous ideal of $R$ is $\star_f$-invertible. Using $\star$-Nagata and Kronecker function rings, we give several different equivalent conditions for $R$ to be a graded Pr\"{u}fer $\star$-multiplication domain. In particular we give new characterizations for a graded integral domain, to be a P$v$MD.