Graded integral domains and Nagata rings

Let R=⨁α∈ΓRαR=⨁α∈ΓRα be an integral domain graded by an arbitrary torsionless grading monoid Γ  . For any f∈Rf∈R, let C(f)C(f) be the ideal of R generated by the homogeneous components of f  , and let N(H)={g∈R|C(g)v=R}N(H)={g∈R|C(g)v=R}. In this paper, we study relationships between the ideal-theoretic properties of RN(H)RN(H) and the homogeneous ideal-theoretic properties of R. For example, we show that R   is a graded Krull domain if and only if RN(H)RN(H) is a Dedekind domain, if and only if RN(H)RN(H) is a PID; and that if R contains a unit of nonzero degree, then R   is a PvMDPvMD if and only if RN(H)RN(H) is a Prüfer domain, if and only if each ideal of RN(H)RN(H) is extended from a homogeneous ideal of R.