On Graded Quasi-Prime Ideals - Hikari Ltd

Int. Journal of Math. Analysis, Vol. 8, 2014, no. 27, 1305 - 1311 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.45149

On Graded Quasi-Prime Ideals Khaldoun Al-Zoubi Department of Mathematics and Statistics Jordan University of Science and Technology P.O. Box 3030, Irbid 22110, Jordan Rashid Abu-Dawwas Department of Mathematics Yarmouk University, Irbid, Jordan c 2014 Khaldoun Al-Zoubi and Rashid Abu-Dawwas. This is an open access Copyright article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Let G be an arbitrary group with identity e and let R be a G-graded commutative ring. In this paper, we introduce the concept of graded quasi-prime ideal and we give a number of results concerning such ideals. In fact, our objective is to investigate graded quasi-prime ideals and examine in particular when graded ideals of R are graded quasi-prime. Also, the relations between graded quasi-prime ideals, graded prime ideals and graded primary ideals of R are studied. Finally, we define a topology on the set of all graded quasi-prime ideals.

Mathematics Subject Classification: 13A02, 16W50 Keywords: graded quasi-prime ideals; graded prime ideals

1

Introduction and Preliminaries

Graded prime ideals in a commutative graded ring have been introduced and studied in [2, 3]. Also, graded primary ideals in a commutative graded ring have been introduced and studied by M. Refai and K. Al-Zoubi in [2]. Here we

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Khaldoun Al-Zoubi and Rashid Abu-Dawwas

study some properties of graded quasi-prime ideals in a commutative graded ring. Before we state some results, let us introduce some notations and terminologies. Let G be a group with identity e and R be a commutative ring. Then R is a G-graded ring if there exist additive subgroups Rg of R such that R = g∈G Rg and Rg Rh ⊆ Rgh for all g, h ∈ G. We denote this by (R, G). The elements of Rg are called homogeneous of degree g where Rg are additive subgroups of R indexed by the elements g ∈ G. If x ∈ R, then x can be written uniquely as g∈G xg , where xg is the component of x in Rg . Moreover, h(R) = g∈G Rg . Let I be an ideal of R. Then I is called graded ideal of (R, G) if I = g∈G (I ∩ Rg ). Thus, if x ∈ I, then x = g∈G xg with xg ∈ I. An ideal of a G-graded ring need not be G-graded. For simplicity, we will denote the graded ring (R, G) by R, see [1]. Let R = ⊕ Rg be a G-graded ring and let I g∈G

be a graded ideal of R. Then the Quotient ring R/I is also a G-graded ring. Indeed, R/I = ⊕ (R/I)g where (R/I)g = {x+I : x ∈ Rg }. Let S ⊆ h(R) be a g∈G

multiplicatively closed subset of R. Then the ring of fraction S −1 R is a graded ring which is called graded ring of fractions. Indeed, S −1 R = ⊕ (S −1 R)g g∈G

where (S −1 R)g = {r/s : r ∈ R, s ∈ S and g = (deg s)−1 (deg r)}. We write h(S −1 R) = ∪ (S −1 R)g . Consider the graded homomorphism η : R → S −1 R g∈G

defined by η(r) = r/1. For any graded ideal I of R the ideal of S −1 R generated by η(I) is denoted by S −1 I. Similar to non graded case, one can prove that S −1 I = {β ∈ S −1 R : β = r/s for r ∈ I and s ∈ S} and that S −1 I = S −1 R if and only if S ∩ I = φ. If J is a graded ideal in S −1 R, then J ∩ R will denote the graded ideal η −1 (J) of R. Moreover, similar to the non graded case one can prove that S −1 (J ∩ R) = J, see [1]. The graded radical of a graded ideal I, denoted by Gr(I), is the set of all n x ∈ R such that for each g ∈ G there exists ng > 0 with xg g ∈ I. Note that, if r is a homogeneous element, then r ∈ Gr(I) if and only if r n ∈ I for some n ∈ N, see [3]. A proper graded ideal P of R is said to be graded prime ideal if whenever r, s ∈ h(R) with rs ∈ P , then either r ∈ P or s ∈ P , see [3]. A proper graded ideal I of R is said to be a graded primary ideal if whenever r, s ∈ h(R) with rs ∈ I, either r ∈ I or s ∈ Gr(I), see [2].

2

Graded quasi-prime ideals

In this section, we define the graded quasi-prime ideals and give some of their basic properties. Definition 2.1. A proper graded ideal P of a graded ring R is said to be a graded quasi-prime if for graded ideals J1 and J2 of R, the inclusion J1 ∩J2 ⊆ P

On graded quasi-prime ideals

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implies that either J1 ⊆ P or J2 ⊆ P. Lemma 2.2. [3, Proposition 1.2] Let R be a G-graded ring and P a proper graded ideal of R. Then I is a graded prime if and only if whenever J1 , J2 are graded ideals of R with J1 J2 ⊆ P , either J1 ⊆ P or J2 ⊆ P. Theorem 2.3. Let R be a G-graded ring and P a graded ideal of R. If P is a graded prime ideal of R, then P is a graded quasi-prime ideal of R. Proof. Let P be a graded prime ideal and J1 , J2 graded ideals of R with J1 ∩ J2 ⊆ P. Hence J1 J2 ⊆ P. By Lemma 2.2, we conclude that either J1 ⊆ P or J2 ⊆ P. Thus P is a graded quasi-prime. Recall that a proper graded ideal I of a graded ring R is said to be graded irreducible ideal if whenever J1 , J2 are graded ideals of R with I = J1 J2 , either I = J1 or I = J2 , see [2]. Also, a graded ring R is called graded noetherian if it is satisfies the ascending chain condition on graded ideals of R, see[1]. Lemma 2.4. [2, Proposition 2.15]. Let R be a G-graded noetherian ring and I a graded irreducible ideal of R. Then I is a graded primary ideal. Theorem 2.5. Let R be a G-graded ring and P a graded ideal of R. If P is a graded quasi-prime ideal of R, then P is a graded irreducible ideal of R. Proof. Assume that P is a graded quasi-prime and J1 , J2 are graded ideals of R such that P = J1 ∩J2 . Since P is a graded quasi-prime ideal and J1 ∩J2 ⊆ P, we have either J1 ⊆ P or J2 ⊆ P and hence either P = J1 or P = J2 . Thus P is a graded irreducible. Corollary 2.6. Let R be a G-graded noetherian ring and P a graded ideal of R. If P is a graded quasi-prime ideal of R, then P is a graded primary ideal of R. Proof. Assume that P is a graded quasi-prime. By Theorem 2.5, P is a graded irreducible. So by Lemma 2.4, we conclude that P is a graded primary. Theorem 2.7. Let R be a G-graded ring and P a graded quasi-prime ideal of R. Then P is a graded prime ideal of R if and only if P = Gr(P ). Proof. (⇒) Let a ∈ Gr(P ) and g ∈ G. Then there exists n ∈ N such that ang ∈ P. Since P is a graded prime, ag ∈ P . Thus a ∈ P , it follows that Gr(P ) ⊆ P. Other side of the inclusion obvious. Therefore P = Gr(P ). (⇐) Let J1 and J2 be graded ideals of R with J1 J2 ⊆ P. By [3, Proposition 2.4], we have J1 ∩ J2 ⊆ Gr(J1 ∩ J2 ) = Gr(J1 J2 ) ⊆ Gr(P ) = P. Since P is a graded quasi-prime, we have either J1 ⊆ P or J2 ⊆ P. By Lemma 2.2, P is a graded prime ideal.

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Theorem 2.8. Let R be a G-graded ring and P a graded quasi-prime ideal of R. If I is a graded ideal contained in P , then P/I is a graded quasi-prime ideal of R/I. Proof. Let J1 and J2 be a graded ideals of R such that (J1 /I) ∩ (J2 /I) ⊆ P/I. Hence (J1 + I)∩ (J2 + I) ⊆ P + I = P. Since P is a graded quasi-prime, either J1 ⊆ P or J2 ⊆ P. Hence either J1 /I ⊆ P/I or J2 /I ⊆ P/I. Thus P/I is a graded quasi-prime. Theorem 2.9. Let R be a G-graded ring and P a proper graded ideal of R. To show that P is a graded quasi-prime, it suffices to show that if whenever rR and sR are graded principle ideals of R with rR ∩ sR ⊆ P, either r ∈ P or s ∈ P. Proof. Let J1 and J2 be a graded ideals of R such that J1 ∩ J2 ⊆ P. Assume tg ∈ J2 . that J1 P. Then there exists r ∈ (J1 ∩ h(R)) − P. Now, let t = g∈G

Then for all g ∈ G, rR ∩ tg R ⊆ J1 ∩ J2 ⊆ P. By our assumption we obtain, tg ∈ P for all g ∈ G. Hence t ∈ P , it follows that J2 ⊆ P. Thus P is a graded quasi-prime. Theorem 2.10. Let R be a G-graded ring and S ⊆ h(R) be a multiplication closed subset of R. If S −1 P is a graded quasi-prime ideal of S −1 R, then S −1 P ∩ R is a graded quasi-prime ideal of R. Proof. Assume that S −1 P is a graded quasi-prime and let J1 , J2 be a graded ideals of R such that J1 ∩ J2 ⊆ S −1 P ∩ R. Hence S −1 J1 ∩ S −1 J2 ⊆ S −1 P. Since S −1 P is a graded quasi-prime, either S −1 J1 ⊆ S −1 P or S −1 J2 ⊆ S −1 P and hence either J1 ⊆ S −1 P ∩ R or J2 ⊆ S −1 P ∩ R. Thus S −1 P ∩ R is a graded quasi-prime ideal. Lemma 2.11. Let R be a G-graded ring, S ⊆ h(R) a multiplication closed subset of R and P a graded ideal of R. If P is a graded primary ideal of R such that Gr(P ) ∩ S = φ, then S −1 P ∩ R = P. Proof. Let m = mg ∈ S −1 P ∩ R. Then for all g ∈ G, there are elements g∈G

nh ∈ P ∩h(R) and s ∈ S such that m1g = nsh . Hence there exists t ∈ S such that stmg = tnh ∈ P. Since P is a graded primary and Gr(P ) ∩ S = φ, mg ∈ P . So m ∈ P, it follows that S −1 P ∩ R ⊆ P. Other side of the inclusion obvious. Thus S −1 P ∩ R = P.

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Theorem 2.12. Let R be a G-graded ring, S ⊆ h(R) a multiplication closed subset of R and P a graded primary ideal of R such that Gr(P ) ∩ S = φ. If P is a graded quasi-prime of R, then S −1 P is a graded quasi-prime ideal of S −1 R. Proof. Assume that P is a graded quasi-prime. Let J1 and J2 be a graded ideals of S −1 R such that J1 ∩ J2 ⊆ S −1 P . Hence (J1 ∩ R) ∩ (J2 ∩ R) ⊆ S −1 P ∩ R. By lemma 2.11, S −1 P ∩ R = P. Since P is a graded quasi-prime, either J1 ∩ R ⊆ P or J2 ∩ R ⊆ P and hence either J1 = (J1 ∩ R)S −1 R ⊆ S −1 P or J2 = (J2 ∩ R)S −1 R ⊆ S −1 P. Thus S −1 P is a graded quasi-prime.

3

Topology on the graded quasi-prime spectrum

Let R be a G-graded ring, we consider qSpecg (R) to be the set of all graded quasi-prime ideals of R. We call qSpecg (R), the graded quasi-prime specrtum of R. For A ⊆ R, let Vg (A) denote the set of all graded quasi-prime of R containing A. Clearly, Vg (A) = Vg (h(A)) where h(A) = h(R) ∩ A. Lemma 3.1. Let R be a G-graded ring. Then (i) If A, B ⊆ R with A ⊆ B, then Vg (B) ⊆ Vg (A). (ii) If A ⊆ R and I is the graded ideal of R generated by h(A), then Vg (A) = Vg (I). (iii) Vg (0) = qSpecg (R) and Vg (1) = φ. (iv) If {Aα : α ∈ Δ} is any family of subsets of R, then Vg ( ∪ Aα ) = α∈Δ

∩ Vg (Aα ).

α∈Δ

(v) Vg (I ∩ J) = Vg (I) ∪ Vg (J) for any two graded ideals I, J of R. Proof. (i)-(iii) Clear. (iv) Let {Aα : α ∈ Δ} be any family of subsets of R. Clearly, Aβ ⊆ ∪ Aα α∈Δ

for all β ∈ Δ. By (i), Vg ( ∪ Aα ) ⊆ Vg (Aβ ) for all β ∈ Δ. Thus Vg ( ∪ Aα ) = ∩ Vg (Aα ). α∈Δ

α∈Δ

α∈Δ

(v) Let I, J be any two graded ideals of R and P ∈ Vg (I ∩ J). Then I ∩ J ⊆ P. Since P is a graded quasi-prime, we have either I ⊆ P or J ⊆ P. i.e., P ∈ Vg (I) or P ∈ Vg (J). Hence Vg (I ∩ J) ⊆ Vg (I) ∪ Vg (J) . Other side of the inclusion obvious. Thus Vg (I ∩ J) = Vg (I) ∪ Vg (J).

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Theorem 3.2. Let R be a G-graded ring and τ = {qSpecg (R) − Vg (A) : A ⊆ R}. Then τ is a topology on qSpecg (R). Proof. Clearly, qSpecg (R) = φ. By Lemma 3.1(iii), we have qSpecg (R) − Vg (0) = φ and qSpecg (R) − Vg (1) = qSpecg (R). Hence φ, qSpecg (R) ∈ τ. Let qSpecg (R) − Vg (A1 ), qSpecg (R) − Vg (A2 ) be any two elements of τ. Then (qSpecg (R)−Vg (A1 ))∩(qSpecg (R)−Vg (A2 )) = qSpecg (R)−(Vg (A1 )∪Vg (A2 )). Let I1 , I2 be the graded ideals of R generated by h(A1 ), h(A2 ), respectively. By Lemma 3.1(ii), Vg (A1 ) = Vg (I1 ) and Vg (A2 ) = Vg (I2 ). By Lemma 3.1(v), Vg (I1 ) ∪ Vg (I2 ) = Vg (I1 ∩ I2 ). Hence (qSpecg (R) − Vg (A1 )) ∩ (qSpecg (R) − Vg (A2 )) = qSpecg (R) − Vg (I1 ∩ I2 ) ∈ τ. Now, let {qSpecg (R) − Vg (Aα ) : α ∈ Δ} be any family of elements of qSpecg (R). Then ∪ (qSpecg (R) − Vg (Aα )) = α∈Δ

qSpecg (R)− ∩ Vg (Aα ). By lemma 3.1(iv), ∪ (qSpecg (R)−Vg (Aα )) = qSpecg (R)− α∈Δ

α∈Δ

Vg ( ∪ Aα ) ∈ τ. Thus τ is a topology on qSpecg (R). α∈Δ

Lemma 3.3. [3, Proposition 2.5] Let R be a G-graded ring and I a graded ideal of R. Then Gr(I) is the intersection of all graded prime ideals of R containing I. Let R be a G-graded ring and r ∈ R, then we denote by opg (r) an open set qSpecg (R) − Vg (r) of qSpecg (R). Theorem 3.4. Let R be a G-graded ring and r, t ∈ h(R). (i) If opg (r) = φ, then r is a nilpotent element of R. (ii) If opg (r) = opg (t), then Gr(Ir ) = Gr(It ), where Ir , It are the graded ideals generated by r and t, respectively. Proof. (i) Let r ∈ h(R) with opg (r) = φ. Hence Vg (r) = qSpecg (R) and so r in every graded quasi-prime ideals of R. Hence r belongs to the intersection of all graded quasi-prime ideals of R. By Theorem 2.3, r belongs to the intersection of all graded prime ideals of R and thus r is a nilpotent element. (ii) Let r, t ∈ h(R) and Ir , It be the graded ideals generated by r and t, respectively such that opg (r) = opg (t). Suppose that P is a graded prime ideal of R with Ir ⊆ P. By Theorem 2.3, P is a graded quasi-prime ideal of R and hence P ∈ Vg (r) = Vg (t) and so It ⊆ P. By Lemma 3.3 , Gr(It ) ⊆ Gr(Ir ). Similarly, we can show that Gr(Ir ) ⊆ Gr(It ). Thus Gr(Ir ) = Gr(It ).


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References [1] C. Nastasescu and F. Van Oystaeyen, Graded ring theory, Mathematical Library 28, North Holand, Amsterdam, (1982). [2] M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish Journal of Mathematics, 28 (2004), 217-229. [3] M. Refai, M. Hailat and S. Obiedat, Graded radicals on graded prime spectra, Far East Journal of Mathematical Sciences I, (2000) 59-73. [4] R.Y. Sharp, Steps in commutative algebra, Cambridge University Press, Cambridge, (1990). Received: May 5, 2014