GRADED INTEGRAL DOMAINS IN WHICH EACH NONZERO HOMOGENEOUS IDEAL IS DIVISORIAL

Abstract

Let Γ be a nonzero commutative cancellative monoid (written additively), R = ⊕ α∈Γ Rα be a Γ-graded integral domain with Rα 6= {0} for all α ∈ Γ, and S(H) = {f ∈ R |C(f) = R}. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if R is integrally closed, then R is a homogeneously divisorial domain if and only if RS(H) is an h-local Prüfer domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-Prüfer domain, (ii) every homogeneous maximal ideal of R is invertible, (iii) each nonzero homogeneous prime ideal of R is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of R has only finitely many minimal prime ideals. We also show that if R is a graded-Noetherian domain, then R is a homogeneously divisorial domain if and only if RS(H) is a divisorial domain of (Krull) dimension one.