DISCRETE VALUATION OVERRINGS OF A GRADED NOETHERIAN DOMAIN

Let R = circle plus(alpha epsilon lambda) R alpha e an integral domain graded by an arbitrary torsionless grading monoid lambda, M a homogeneous maximal ideal of R and S(H) = R backslash Up epsilon h- Spec(R) P. We show that R is a graded Noetherian domain with h-dim(R) = 1 if and only if R-s(H) (is a one_) dimensional Noetherian domain. We then use this result to prove a graded Noetherian domain analogue of the KrullAkizuki theorem. We prove that, if R is a gr-valuation ring, then R-M is a valuation domain, dim (R-M) = h-dim(R) and R-M is a discrete valuation ring if and only if R is discrete as a gr-valuation ring. We also prove that, if {P-i} is a chain of homogeneous prime ideals of a graded Noetherian domain R, then there exists a discrete valuation overring of R which has a chain of prime ideals lying over {P-z}.