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Some properties of gr-multiplication ideals

In this paper, we study some of the properties of gr-multiplication ideals in a graded ring R. We first characterize finitely generated gr-multiplication ideals and then give a characterization of gr-multiplication ideals by using the gr-localization of R. Finally we determine the set of gr-P primary ideals of R when P is a gr-multiplication gr-prime ideal of R. | Turk J Math 33 (2009) , 205 – 213. ¨ ITAK ˙ c TUB doi:10.3906/mat-0801-16 Some properties of gr-multiplication ideals Hani A. Khashan Abstract In this paper, we study some of the properties of gr-multiplication ideals in a graded ring R . We first characterize finitely generated gr-multiplication ideals and then give a characterization of gr-multiplication ideals by using the gr-localization of R . Finally we determine the set of gr- P -primary ideals of R when P is a gr-multiplication gr-prime ideal of R . Key Words: Graded Rings, Graded Ideals, Gr-primary Ideals and Gr-multiplication Ideals. 1. Introduction Let G be a group. A ring (R, G) is called a G -graded ring if there exists a family {Rg : g ∈ G} of Rg and Rg Rh ⊆ Rgh for each g and h in G . For simplicity, we additive subgroups of R such that R = g∈G will denote the graded ring (R, G) by R . An element of a graded ring R is called homogeneous if it belongs Rg and this set of homogeneous elements is denoted by h(R). If x ∈ Rg for some g ∈ G , then we say to g∈G that x is of degree g . A graded ideal I of a graded ring R is an ideal verifying I = (I ∩ Rg ) = g∈G Equivalently, I is graded in R if and only if I has a homogeneous set of generators. If R = R = g∈G Ig . g∈G Rg and g∈G R g are two graded rings, then a mapping η : R → R with η(1R ) = 1R is called a gr-homomorphism if η(Rg ) ⊆ R g for all g ∈ G . A graded ideal P of a graded ring R is called gr-prime if whenever x, y ∈ h(R) with xy ∈ P , then x ∈ P or y ∈ P . A graded ideal M of a graded ring R is called gr-maximal if it is maximal in the lattice of graded ideals of R . A graded ring R is called a gr-local ring if it has unique gr-maximal ideal. Let R be a graded ring and let S ⊆ h(R) be a multiplicatively closed subset of R . Then the ring of −1 fractions S −1 R is a graded ring which is called the gr-ring of fractions. Indeed, S −1 R = S R g where g∈G r −1 S R g= : r ∈ R , s ∈ S and g = (deg s)−1 .