Characterizations of graded Prüfer $\star$-multiplication domains

Let $R=\bigoplus_{α\inΓ}R_α$ be a graded integral domain graded by an arbitrary grading torsionless monoid $Γ$, 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 Prüfer $\star$-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 Prüfer $\star$-multiplication domain. In particular we give new characterizations for a graded integral domain, to be a P$v$MD.

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