Graded-valuation domains. Issue 9 (2nd September 2017)

• Record Type: Journal Article
• Authors: Anderson, D. D.; Anderson, David F.; Chang, Gyu Whan
• Is Part Of: Communications in algebra. Volume 45:Issue 9(2017)
• Page Start: 4018
• Page End: 4029
• Publication Date: 2017-09-02
• Subjects: Graded integral domain -- graded-valuation (domain) -- twisted semigroup ring; 13A02 -- 13A15 -- 13A18 -- 20M25; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1254784
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let Γ be a torsionless grading monoid, R = ⊕ α ∈ Γ R α a Γ -graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R 0, and G 0 = Γ ∩− Γ the group of units of Γ . We say that R is a graded-valuation domain if either x ∈ R or x −1 ∈ R for every nonzero homogeneous element x ∈ R H . In this paper, we show that R is a graded-valuation domain if and only if Γ is a valuation monoid, R α = Kx for every 0≠ x ∈ R α whenever α is not a unit of Γ, and T = ⊕ α ∈ G 0 R α is a graded-valuation domain. Let R = K γ [ X ; Γ ] be a twisted semigroup ring of Γ over K, C a totally ordered (additive) abelian group, C ′ a subgroup of C, μ : K → C ′ ∪{∞} a valuation, φ : Γ → C a function such that C ′ ∪ φ ( Γ ) generates C, and v : R → C ∪{∞} the function defined byv ( ∑ a α X α ) = inf { μ ( a α ) + φ ( α ) } for every∑ a α X α ∈ R . We show that v is a valuation if and only if μ ( γ ( a, b ))+ φ ( a + b ) = φ ( a )+ φ ( b ) for every a, b ∈ Γ .