Graded Prüfer domains. Issue 2 (1st February 2018)

• Record Type: Journal Article
• Authors: Anderson, David F.; Chang, Gyu Whan; Zafrullah, Muhammad
• Is Part Of: Communications in algebra. Volume 46:Issue 2(2018)
• Page Start: 792
• Page End: 809
• Publication Date: 2018-02-01
• Subjects: Graded Prüfer domain -- homogeneous overring -- monoid domain -- Nagata ring -- torsionless grading monoid; 13A02 -- 13A15 -- 13A18 -- 13B25 -- 13F05 -- 13G05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1327595
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let R = ⊕ α ∈ Γ R α be an integral domain graded by an arbitrary torsionless grading monoid Γ . We say that R is a graded Prüfer domain if every nonzero finitely generated homogeneous ideal of R is invertible. In this paper, we generalize several results on Prüfer domains to graded Prüfer domains. Among other things, we show that (i) R is a graded Prüfer domain if and only if every homogeneous overring of R is t -flat over R, if and only if every homogeneous overring of R is integrally closed; (ii) if every homogeneous overring of R is a quotient ring of R, then R is a graded Prüfer domain; (iii) if A ⊆ B is an extension of integral domains, Γ ∩− Γ = {0}, Γ * = Γ ∖{0}, and R = A + B [ Γ *], then R is a graded Prüfer domain if and only if A is a Prüfer domain, B is the quotient field of A, and Γ is a valuation monoid; and (iv) if S is a torsionless grading monoid, then the monoid domain R [ S ] is a graded Prüfer domain if and only if either R is a graded Prüfer domain and S is a group or every nonzero homogeneous element of R is a unit and S is a valuation monoid.