Journal article
Graded-valuation domains
Communications in Algebra, Vol.45(9), pp.4018-4029
2017
DOI: 10.1080/00927872.2016.1254784
Abstract
Let Γ be a torsionless grading monoid, R = ⊕α∈ΓRα a Γ-graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R0, and G0 = 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∈RH. In this paper, we show that R is a graded-valuation domain if and only if Γ is a valuation monoid, Rα = Kx for every 0≠x∈Rα whenever α is not a unit of Γ, and T = ⊕α∈G0Rα is a graded-valuation domain. Let R = Kγ[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 by v(∑aαXα) for every ∑aαXα. 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. © 2017 Taylor & Francis.
Details
- Title: Subtitle
- Graded-valuation domains
- Creators
- D.D. Anderson - University of IowaD.F. Anderson - University of Tennessee at KnoxvilleG.W. Chang - Incheon National University
- Resource Type
- Journal article
- Publication Details
- Communications in Algebra, Vol.45(9), pp.4018-4029
- Publisher
- Taylor and Francis Inc.
- DOI
- 10.1080/00927872.2016.1254784
- ISSN
- 0092-7872
- Language
- English
- Date published
- 2017
- Academic Unit
- Mathematics
- Record Identifier
- 9984230627702771
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