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On graded weakly prime ideals

Here we study the graded weakly prime ideals of a G-graded commutative ring. A number of results concerning graded weakly prime ideals are given. For example, we give some characterizations of graded weakly prime ideals and their homogeneous components. | Turk J Math 30 (2006) , 351 – 358. ¨ ITAK ˙ c TUB On Graded Weakly Prime Ideals Shahabaddin Ebrahimi Atani Abstract Let G be an arbitrary group with identity e, and let R be a G-graded commutative ring. Weakly prime ideals in a commutative ring with non-zero identity have been introduced and studied in [1]. Here we study the graded weakly prime ideals of a G-graded commutative ring. A number of results concerning graded weakly prime ideals are given. For example, we give some characterizations of graded weakly prime ideals and their homogeneous components. Key Words: Graded rings, Graded weakly prime ideals. 1. Introduction Weakly prime ideals in a commutative ring with non-zero identity have been introduced and studied by D. D. Anderson and E. Smith in [1]. Also, weakly primary ideals in a commutative ring with non-zero identity have been introduced and studied in [2]. Here we study the graded weakly prime ideals of a G-graded commutative ring. The purpose of this paper is to explore some basic facts of these class of ideals. Various properties of graded weakly prime ideals are considered. First, we show that if P is a graded weakly prime ideal, then for each g ∈ G, either Pg is a prime subgroup of Rg or Pg2 = 0. Also, we show that if P and Q are graded weakly prime ideals such that Pg and Qh are not prime for all g, h ∈ G respectively, then Grad(P ) = Grad(Q) = Grad(0) and P + Q is a graded weakly prime ideal of G(R). Next, we give some characterizations of graded weakly prime ideals and their homogeneous components (see sec. 2). Before we state some results let us introduce some notation and terminology. Let G be an arbitrary group with identity e. By a G-graded commutative ring we mean a 351 ATANI commutative ring R with non-zero identity together with a direct sum decomposition (as an additive group) R = ⊕g∈G Rg with the property that Rg Rh ⊆ Rgh for all g, h ∈ G; here Rg Rh denotes the additive subgroup of R consisting of all finite sums of elements