Graded integral domains in which each nonzero homogeneous t-ideal is divisorial

Let [Formula: see text] be a nonzero commutative cancellative monoid (written additively), [Formula: see text] be a [Formula: see text]-graded integral domain with [Formula: see text] for all [Formula: see text], and [Formula: see text]. In this paper, we study graded integral domains in which each nonzero homogeneous [Formula: see text]-ideal (respectively, homogeneous [Formula: see text]-ideal) is divisorial. Among other things, we show that if [Formula: see text] is integrally closed, then [Formula: see text] is a P[Formula: see text]MD in which each nonzero homogeneous [Formula: see text]-ideal is divisorial if and only if each nonzero ideal of [Formula: see text] is divisorial, if and only if each nonzero homogeneous [Formula: see text]-ideal of [Formula: see text] is divisorial.

On the union of graded prime ideals

In this paper we investigate graded compactly packed rings, which is defined as; if any graded ideal I of R is contained in the union of a family of graded prime ideals of R, then I is actually contained in one of the graded prime ideals of the family. We give some characterizations of graded compactly packed rings. Further, we examine this property on h – Spec(R). We also define a generalization of graded compactly packed rings, the graded coprimely packed rings. We show that R is a graded compactly packed ring if and only if R is a graded coprimely packed ring whenever R be a graded integral domain and h – dim R = 1.

WHEN ARE GRADED INTEGRAL DOMAINS ALMOST GCD-DOMAINS

Let R = ⨁α∈Γ Rα be a (Γ-)graded integral domain and let H be the multiplicatively closed set of nonzero homogeneous elements of R. In this paper, we introduce the concepts of graded almost GCD-domains (graded AGCD-domain) and graded almost Prüfer v-multiplication domains (graded AP v MD ). Among other things, we show that if R is integrally closed, then (1) H is an almost lcm splitting set of R if and only if R is a graded AGCD-domain and (2) R is a graded AP v MD if and only if R is a P v MD . We also give an example of a (non-integrally closed) graded AGCD-domain (respectively, graded AP v MD ) that is not an almost GCD-domain (respectively, almost Prüfer v-multiplication domain.

A Note on Buchsbaum Rings and Localizations of Graded Domains

Let R = ⊕i ≧0Ri be a graded integral domain, and let p ∈ Proj (R) be a homogeneous, relevant prime ideal. Let R(p) = {r/t| r ∈ Ri, t ∈ Ri\p} be the geometric local ring at p and let Rp = {r/t| r ∈ R, t ∈ R\p} be the arithmetic local ring at p. Under the mild restriction that there exists an element r1 ∈ R1\p, W. E. Kuan [2], Theorem 2, showed that r1 is transcendental over R(p) andwhere S is the multiplicative system R\p. It is also demonstrated in [2] that R(p) is normal (regular) if and only if Rp is normal (regular). By looking more closely at the relationship between R(p) and R(p), we extend this result to Cohen-Macaulay (abbreviated C M.) and Gorenstein rings.