Graded integral domains which are UMT-domains

Abstract

ABSTRACTLet Γ be a torsionless commutative cancellative monoid, 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 RQ 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 RQ is a quasi-Prüfer domain for each homogeneous maximal t-ideal Q of R and (ii) R is a PvMD if and only if every nonzero finitely generated homogeneous ideal of R is t-invertible, if and only if RQ 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 ΓS is a valuation monoid for all maximal t-ideals S of Γ. Hence, D[Γ] is a PvMD if and only if D is a PvMD and Γ is a Prüfer v-multiplication semigroup.