On the union of graded prime ideals

Abstract

Abstract:In this paper we investigate graded compactly packed rings, which is de ned 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 de ne 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 = .

On Graded 2-Absorbing Second Submodules of Graded Modules over Graded Commutative Rings

“…The notion of graded prime ideals was introduced in [24] and studied in [12,23,25]. A proper graded ideal P of R is said to be a graded prime ideal if whenever rs ∈ P , we have r ∈ P or s ∈ P, where r, s ∈ h(R).…”

The Graded Classical Prime Spectrum with the Zariski Topology as a Notherian Topological Space

“…Let G be a group and R be a G-graded commutative ring. A proper graded ideal I of R is said to be a graded prime ideal if whenever rs ∈ I, we have r ∈ I or s ∈ I, where r, s ∈ h(R), (for more details see [36,37,38]). Recall that the spectrum Spec g (R) of a graded ring R consists of all graded prime ideals of R. For every graded ideal I of R, we set V g R (I) = {P ∈ Spec g (R)|I ⊆ P }.…”

The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings

“…The concept of graded prime ideal was introduced by M. R e f a i, M. H a i l a t and S. O b i e d a t in [10] and studied in [1,8,11].…”