Integral Closure of Graded Integral Domains

Park, Mi Hee

doi:10.1080/00927870701509511

Cited by 14 publications

(4 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

mentioning

confidence: 56%

On Graded Krull Overrings of a Graded Noetherian Domain

Lee¹,

Park²

2012

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

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

show abstract

mentioning

confidence: 56%

On Graded Krull Overrings of a Graded Noetherian Domain

Lee¹,

Park²

2012

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…(1) ⇒ Assume (1). Let T be a homogeneous overring of R. Then by [19,Corollary 1.6], T is a h-Noetherian domain and h-dim(T ) ≤ 1. Hence, T is a gGD domain.…”

Section: Graded Going Down Domainsmentioning

confidence: 99%

On graded pseudo-valuation domains

Sahandi

2021

Communications in Algebra

View full text Add to dashboard Cite

Let Γ be a torsionless commutative cancellative monoid and R = α∈Γ Rα be a Γ-graded integral domain. In this paper, we introduce the notion of graded going-down domains. Among other things, we provide an equivalent condition for graded-Prüfer domains in terms of graded going-down and graded finite-conductor domains. We also characterize graded going-down domains by means of graded divided domains. As an application, we show that the graded going-down property is stable under factor domains.

show abstract

“…This action extends to the normalization S of S (see e.g. [P,Theorem 2.10]). The Poisson structure of S also extends to S (cf.…”

Section: Non-normal Surfacesmentioning

confidence: 99%

Four-dimensional conical symplectic hypersurfaces

Yamagishi

2019

Preprint

View full text Add to dashboard Cite

We show that every indecomposable conical symplectic hypersurface of dimension four is isomorphic to the known one, namely, the Slodowy slice X n which is transversal to the nilpotent orbit of Jordan type [2n − 2, 1, 1] in the nilpotent cone of sp 2n for some n ≥ 2. In the appendix written by Yoshinori Namikawa, conical symplectic varieties of dimension two are classified by using contact Fano orbifolds. Conical symplectic varieties as graded Poisson algebrasAn affine symplectic varietya finitely generated graded C-algebra with R 0 = C and the algebraic symplectic form ω on the regular part X reg is homogeneous with respect to the C * -action λ : C * × X → X coming from the grading on R, that is, there exists an integer s such that λ(t) * (ω) = t s ω for any t ∈ C * . By [LNSvS, Lemma 2.2], the weight s is positive.Let X = Spec R be a conical symplectic variety. The symplectic form ω is regarded as an isomorphism ω : T Xreg → Ω 1Xreg . Then the structure sheaf O Xreg admits a natural Poisson structure defined by {f, g} = ω(ω −1 (df ), ω −1 (dg)) for any sections f, g of O Xreg . Namely, the pairingis a skew C-bilinear form satisfying the Leibniz rule {f g, h} = f {g, h} + g{f, h} and the Jacobi identity {f, {g, h}} + {g, {h, f }} + {h, {g, f }} = 0 for any sections f, g and h of O Xreg .

show abstract

Integral Closure of Graded Integral Domains

Cited by 14 publications

References 15 publications

On Graded Krull Overrings of a Graded Noetherian Domain

On Graded Krull Overrings of a Graded Noetherian Domain

On graded pseudo-valuation domains

Four-dimensional conical symplectic hypersurfaces

Contact Info

Product

Resources

About