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