Graded integral domains which are UMT-domains. Issue 6 (3rd June 2018)

• Record Type: Journal Article
• Authors: Chang, Gyu Whan; Sahandi, Parviz
• Is Part Of: Communications in algebra. Volume 46:Issue 6(2018)
• Page Start: 2742
• Page End: 2752
• Publication Date: 2018-06-03
• Subjects: Graded integral domain -- monoid domain -- PvMD -- UMT-domain -- valuation domain; 13A02 -- 13A15 -- 13F05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1399406
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let Γ be a torsionless commutative cancellative monoid, R = ⊕ α ∈ Γ R α be a Γ -graded integral domain, and H be the set of nonzero homogeneous elements of R . In this paper, we show that if Q is a maximal t -ideal of R with Q ∩ H = ∅, then R Q is a valuation domain. We then use this result to give simple proofs of the facts that (i) R is a UMT-domain if and only if R Q is a quasi-Prüfer domain for each homogeneous maximal t -ideal Q of R and (ii) R is a P v MD if and only if every nonzero finitely generated homogeneous ideal of R is t -invertible, if and only if R Q is a valuation domain for all homogeneous maximal t -ideals Q of R . Let D [ Γ ] be the monoid domain of Γ over an integral domain D . We also show that D [ Γ ] is a UMT-domain if and only if D is a UMT-domain and the integral closure of Γ S is a valuation monoid for all maximal t -ideals S of Γ . Hence, D [ Γ ] is a P v MD if and only if D is a P v MD and Γ is a Prüfer v -multiplication semigroup.