On Graded Krull Overrings of a Graded Noetherian Domain

Abstract

Abstract. Let R be a graded Noetherian domain and A a graded Krull overring of R. We show that if h-dim R ≤ 2, then A is a graded Noetherian domain with h-dim A ≤ 2. This is a generalization of the well-known theorem that a Krull overring of a Noetherian domain with dimension ≤ 2 is also a Noetherian domain with dimension ≤ 2.Let R = ⊕ γ∈Γ R γ be a commutative ring with identity graded by an arbitrary torsionless grading monoid Γ. That is, Γ is a commutative cancellative additive monoid with torsion-free quotient group G = ⟨Γ⟩. A torsionless grading monoid can be given a total order compatible with the monoid operation [2, Corollary 3.4]. This is used in many referenced materials.This paper continues the study of graded Noetherian rings begun in [7,8]. In particular, we will show that, if R is a graded Noetherian domain with h-dim R ≤ 2, then each graded Krull overring of R is graded Noetherian with h-dimension ≤ 2. This is a graded version of the following Heinzer Theorem The graded ring R is called a graded Noetherian ring if R satisfies the ascending chain condition on homogeneous ideals, or equivalently, if each homogeneous (prime) ideal of R is finitely generated. The h-height of a homogeneous prime ideal P (denoted by h-ht P ) is defined to be the supremum of the lengths of chains of homogeneous prime ideals descending from P and the h-dimension of R (denoted by h-dim R) is defined to be sup { h-ht P | P is a homogeneous prime ideal of R}.If R is a Γ-graded integral domain, then the set S of nonzero homogeneous elements of R is a multiplicatively closed set. The quotient ring R S = ⊕ γ∈G (R S ) γ is a G-graded ring, where (R S ) γ = { a b | a ∈ R α , b ∈ R β \ {0}, and α − β = γ} for each γ ∈ G. It is called the homogeneous quotient field