On the union of graded prime ideals

Rabia Nagehan Uregen 1 , Unsal Tekir 2 , and Kursat Hakan Oral 3
  • 1 Yildiz Technical University Graduate School Of Natural and Applied Sciences, 34349 Istanbul-Turkey
  • 2 Marmara University, Department of Mathematics, 34722 Istanbul-Turkey
  • 3 Yildiz Technical University, Department of Mathematics, 34220 Istanbul-Turkey
Rabia Nagehan Uregen, Unsal Tekir and Kursat Hakan Oral

Abstract

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 hSpec(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.

1 Introduction

Throughout this paper, R will be a commutative ring with identity 1R. R is a ℤ-graded ring if there exist additive subgroups Rg of R indexed by the elements g ∈ ℤ such that and R=gRg satisfies RgRhRgh for all g,h ∈ ℤ. The elements of Rg are homogeneous elements of R of degree g, and all homogeneous elements of the ring R are denoted by h(R), i.e.h(R)=gRg. If ab = 0 implies a = 0 or b = 0 for nonzero homogeneous elements a,bh(R), then R is called graded integral domain. A subset S of h(R) is called homogeneous multiplicatively closed subset or shortly multiplicatively closed if a,bS implies abS. Then S –1R, the ring of fraction is a graded ring with S1R=g(S1R)g; where, (S1R)g={rs:rR,  sS  and  g=(degr)(degs)}. Let I be an ideal of R. If I=gIg where Ig = IRg, then I is called graded ideal of R. A graded ideal I is a graded prime ideal of R if I/= R and whenever abI, then either aI or bI, for a, bh(R). The set of all graded prime ideals is denoted by hSpec(R). The maximal elements with respect to the inclusion in the set of all proper graded ideals are graded maximal ideals and the set of all graded maximal ideals is denoted by h - Max(R). A graded ring with finite number of graded maximal ideals is a graded semilocal ring. Further the set of all minimal graded prime ideals is denoted by hMin(R). The graded height of a graded prime ideal P denoted by h - htP, is defined as the length of the longest chain of graded prime ideals contained in P. The Krull dimension of a graded ring R is denoted by h - dim(R) and defined as h – dim(R) = max {h - htP|Ph - Spec(R)} [1].

The finite union of graded prime submodules are studied in [2]. For more details of graded prime submodules refer [3]. Moreover, the finite union of ideals are studied by Quartororo and Butts with the notion of u – ideal in [4]. With these motivations we investigate some properties of the finite union of graded ideals in Section 2. For this, we define graded u – ideal as follow: A graded ideal I is graded u- ideal if it is contained in the finite union of a family of graded ideals of R, then I is actually contained in one of the graded ideal of the family.

In Section 3, we examine some properties of graded compactly packed rings. Compactly packed rings have been studied by various authors, see, for example, [57]. The ring R is compactly packed if for any ideal I of R, IαΔPα where {Pα}α ∈ ∆ is a family of prime ideals of R with the index set ∆, then IPa for some α ∈ ∆. This concept was pointed out by C. Reis and Viswanathan in [5]. They also characterized on Noetherian rings that are compactly packed by prime ideal of R if and only if every prime ideal is the radical of a principal ideal in R. After this work, Smith [7] shows for this property that the ring need not be a Noetherian ring. Moreover Pakala and Shores [5] showed that for a compactly packed Noetherian ring the maximal ideal is the radical of a principal ideal. In [7], Principal Ideal Theorem of Krull was proven for graded rings and using this theorem we show that if R is a graded compactly packed ring, then h − dim R ≤ 1 whenever R be a graded Noetherian ring.

In Section 4, we define graded coprimely packed rings that Erdoğdu [8] defined coprimely packed rings as a generalization of compactly packed rings. An ideal I is coprimely packed if for an index set ∆ and α ∈ ∆, I +Pα = R implies IαΔPα where {Pα}α∈∆ is a family of prime ideals of R. If every ideal of R is coprimely packed, then R is a coprimely packed ring. For the studies about coprimely packed rings the reader is referred to [810]. Finally we show that every graded compactly packed ring is a graded coprimely ring. Additionally, we also show that R is graded coprimely packed ring if and only if R is coprimely packed ring by hMax(R).

2 Finite union of graded ideals

Definition 1. Let R be a graded ring and I be a graded ideal of R. Then we say that I is a graded u-ideal if for any family of graded ideals{Ai}i=1n, Ii=1nAiimplies IAjfor some j = 1, 2, ..., n. A graded ring R is graded u-ring if any graded ideal is a graded u – ideal.

Proposition 1. Let R be a graded ring and I be a graded ideal of R. Then the following conditions are equivalent;

(i) R is a graded u –ring,

(ii) Each finitely generated graded ideal of R is graded uideal,

(iii) IfI=i=1nAiis finitely generated, then I = Aj for some j,

(iv) If I=i=1nAi, then I = Aj for some j.

Proof. (i) ⇒ (ii) is trivial.

(ii) ⇒ (iii) Since I=i=1nAi we get Ii=1nAi and so IAj for some j = {1, 2, ..., n}. Then i=1nAi=IAji=1nAi and so I = Aj.

(iii) ⇒ (iv) Suppose that I=i=1nAi and assume that I/ = Aj for all j ∈ {1, 2, ..., n}. Then there exists an element ajI\ Aj and set J = (a1, ..., an). Then J=JI=J(i=1nAi)=i=1n(JAi) by (iii) we have J = JAj for some j ∈ {1, 2, ..., n}. Hence JAj, which is a contradiction.

(iv) ⇒ (i) It follows from, Ii=1nAi implies I=i=1n(IAi). □

Proposition 2. Every homomorphic image of a graded u-ring is a graded u-ring.

Proof. It is explicit. □

Proposition 3. If R is a graded uring, then S−1R is a graded uring.

Proof. It follows from [[2], Proposition 2.7]. □

3 Graded compactly packed rings

Definition 2. Let R be a graded ring and ∆ an index set. If for any graded ideal I and any family of graded prime ideals {Pα}α∈∆, IαΔPα implies IPβfor some β ∈ ∆, then R is a graded compactly packed ring with hSpec(R).

Proposition 4. Every homomorphic image of a graded compactly packed ring is a graded compactly packed ring.

Proof. Let R be a graded compactly packed ring and S be any graded ring. Let f : RS be an epimorphism. Assume that {Pα'}αΔ be a family of graded prime ideals of S and I ′ be a graded ideal of S such that IαΔPα. Since f is an epimorphism, there exist graded prime ideals Pα and graded ideal I of R such that KerfPα, KerfI and f(Pα)=Pα, f (I) = I0. It follows that f(I)αΔf(Pα)f(αΔPα). Therefore IαΔPα, and so IPβ for some β ∈∆. Thus I=f(I)f(Pβ)=Pβ for some β ∈ ∆. □

Now recall the following well known Lemma.

Lemma 1. [3, Lemma 2.1] Let R be a graded ring, ah(R) and I, J be graded ideals of R. Then aR, I + J and IJ are graded ideals.

Note that the graded ideal aR is denoted by (a).

Theorem 1. Let R be a graded ring, I be a graded ideal and Sh(R) be a multiplicatively closed subset. Then the set

ψ={JJ  is  a  graded  ideal  of  R,  SJ=,  IJ}

has a maximal element and such maximal elements are graded prime ideals of R.

Proof. Since I ∈ ψ, we get ψ/ = ∅. The set ψ is partially ordered set with respect to set inclusion "⊆". Now let ∆ be a totally ordered subset of ψ. Then J=JΔJ is an ideal of R and Jg=JRg=(JΔJ)Rg=JΔ(JRg)=JΔJg. Thus ℑ = ⊕Jg and so ℑ; is graded ideal. Now let P be a maximal element of ψ and a,bh(R) such that aP and bP. Then PP + (a) and P + (a) is a graded ideal. Therefore (P + (a)) ⋂ S/ = ∅ and so there exist sS such that s = x + ar for some xP,rR. Similarly there exist s′ ∈ S such that s′ = x′ + br′ for some x′ ∈ P,r′ ∈ R. Then ss′ = (x + ar)(x′ + br′) = xx′ + arx′ + brx + abrr′. Then abrr′ ∈ P and so abP. Hence P is a graded prime ideal. □

Theorem 2. Let R be a graded ring. Then the following are equivalent:

(i) R is a graded compactly packed ring.

(ii) For every graded prime ideal P, PαΔPα implies PPβfor some β ∈ ∆.

(iii) Every graded prime ideal of R is the radical of a graded principal ideal in R.

Proof. (i) ⇒ (ii) It follows from the definition of graded compactly packed ring.

(ii) ⇒ (iii) Suppose that P is a graded prime ideal of R. Assume that P is not the radical of a graded principal ideal of R. Then we get (r)/=P for all rPh(R). Hence there is a prime ideal Pr such that rPr and PPr for all rPh(R). Further we have PrPh(R)Pr. Then by (ii) PPr′, r′ ∈ Ph(R) which is a contradiction.

(iii) ⇒ (i) Suppose thatIαΔPα. Since h(R)\(αΔPα) is a graded multiplicatively closed subset, there exists a graded prime ideal P such that IP and PαΔPα. Suppose that P=r for some rh(R). Then we get that rαΔPα and rαΔPα. Hence there exists β ∈ ∆ such that rPβ. Therefore IP=rPβ.

Theorem 3. (Principal Ideal Theorem, [[1], Theorem 3.5]) Let x be a nonunit homogeneous element in a graded Noetherian ring R and let P be a graded prime ideal minimal over (x). Then hhtP ≤ 1

Theorem 4. Let R be a graded Noetherian ring. If R is a graded compactly packed ring, then h – dim R ≤ 1.

Proof. Suppose that R is a graded compactly packed ring. Then there exists an rh(R) such that I=P where I = (r) for any graded prime ideal P of R by Theorem 2. From Principal Ideal Theorem we have hhtP ≤ 1. Thus h – dim R ≤ 1. □

Theorem 5. Let R be a graded ring, I be a graded ideal and P be a graded prime ideal such that IP. Then the following are equivalent:

(i) P is a graded minimal prime ideal of I,

(ii) h(R)\P is a graded multiplicatively closed subset that is maximal with respect to missing I,

(iii) For each xPh(R), there is a yh(R)\Pand a nonnegative integer i such that yxiI.

Proof. (i) ⇒ (ii) Suppose that P is a graded minimal prime ideal of I . If we set S = h(R) P, then S is a graded multiplicatively closed subset and there exists a maximal element in the set of graded ideals containing I and disjoint from S. Assume Q is maximal then Q is graded prime ideal by Theorem 1. Since P is minimal, P = Q and so S is maximal with respect to missing I.

(ii) ⇒ (iii) Let 0/= x ∈ P ⋂ h(R) and S = {yxi| yh(R) P, i = 0, 1, 2, ...}. Then h(R)\PS. Since h(R)\ P is maximal, there exist an element yh(R)\ P and i nonnegative integer such that yxiI.

(iii) ⇒ (i) Assume that IQP, where Q is graded prime ideal. If there exists xP\Q where xh(R), then there exist an element yh(R)\P such that yxiI for some i = 0, 1, 2, .... Therefore yxiQ, yQ. Thus xiQ. It is a contradiction.

Recall that a graded ring R is reduced if its nilradical is zero, i.e.

PhSpec(R)P=(0).

Corollary 1.

If R is a graded reduced ring and P is a graded prime ideal of R, then P is a graded minimal prime ideal of R if and only if for each xPh(R) there exists some yh(R)\ P such that xy = 0.

Theorem 6.

let R be a graded ring and hMin(R) = {Pα}. If Pαα=βPβ for each α, then hMin(R) is finite.

Proof. Without loss of generality, assume that R is a graded reduced ring. Then for PhMinR, Rp is a graded field. Let R=RPα. Then ψ : RR′, r ↦ θα(r) where θα : RRPα be the canonical homomorphism. If hMinR is infinite, then RPα is a proper ideal in R ′. Let 𝔐 be a graded maximal ideal of R′ containing the idealRPα. Then 𝔐 ⋂ R contains a graded minimal prime ideal Pγ of R. Choose aPγh(R) where aβ=γPβ. Since R is a graded subring of R ′, a is identified with θα(a). For β/ = γ, θβ(a) is a unit in the graded ring RPβ. Now define an element b = {bβ} ∈ R′ where bβ = θβ(a) –1 for β/= γ and bβ = 0 if β = γ. Then we have ab ∈ 𝔐 where only the γ component is 0 and other components are identity. And so 𝔐 contains the identity of R ′, since RPα𝔐 . This gives us a contradiction. □

Corollary 2. Let R be a graded u-ring and hMin(R) = {Pα}. ThenPαα=βPβfor each α if and only if hMin(R) is finite.

Now we will investigate the graded compactly packing property on graded spectrum of a graded ring and refer to this as the (*) property. The topology of graded spectrum was studied in [11]. For a graded ring R its graded spectrum, hSpec(R), is a topology with the closed sets VGR(I)={PhSpec(R)  |  IP} where I is a graded ideal of R. This topology is called Zariski topology. For any homogeneous element rh(R) define Dr = {PhSpec(R)| rP}, and so the set {Dr| rh(R)} is a basis for the Zariski topology on hSpec(R) [11], Theorem 2.3]. Further Dr is quasi-compact for all rh(R).

Definition 3. Let R be a graded ring, Λ an index set and r, sαh(R)\{0} for all α ∈ Λ. Then we say that R has property (*) DrαΛDsαimplies DrDsβ for some βΛ.

Theorem 7. Let R be a graded ring. If R satisfies property (*) then R has at most two graded maximal ideals.

Proof. Suppose that R has property (*) and assume that 𝔐1, 𝔐2, 𝔐3 are three distinct graded maximal ideals of R. Then we have a ∈ 𝔐1h(R) and b ∈ 𝔐2h(R) such that a + b = 1R. Now let c ∈ (𝔐3h(R))\(𝔐1 ⋃ 𝔐2). Since c = ca + cb, we get DcDacDbc. Since R satisfies (*) property we get DcDac or DcDbc. Both of them is a contradiction. □

Corollary 3. Let R be a graded ring and every nonzero graded prime ideal is a graded maximal ideal. Then R has at most two nonzero graded prime ideals if and only if R satisfies (*) property.

Proof. Suppose that R has at most two nonzero graded prime ideals. If r is a nonzero nonunit homogeneous element of R then Dr\ {(0)} is an empty set or single point set. Then R satisfies (*) property. For the converse, if R satisfies (*) property then by Theorem 7, R has at most two graded maximal ideals. Therefore, this completes the proof. □

4 Graded coprimely packed rings

Definition 4. Let R be a graded ring and I be a graded ideal. I is said to be graded coprimely packed ring if I + Pα = R where Pα (α ∈ ∆) are graded prime ideals of R; thenIαΔPα. If every graded ideal of R is a graded coprimely packed ring, then R is a graded coprimely packed ring by hSpec(R).

Proposition 5. Every homomorphic image of a graded coprimely packed ring is a graded coprimely packed ring.

Proof. Let R be a graded coprimely packed ring and S be a ring. Let f : RS be an epimorphism. Assume that j be a graded ideal of S and {Pα'}αΔ be a family of graded prime ideals of S such that J+Pα'=S. Since f is an epimorphism, there exists a graded ideal I and graded prime ideals Pα of R such that KerfI, KerfPα, f(I) = J and f(Pα)=Pα'. Thus we obtain J+Pα'=f(I+Pα)=f(R)=S. To show that I + Pα = R, let rR. Then f(r) ∈ f(R) = f(I + Pα). Then there exists mI + Pα such that f(r) = f(m), that is rmKer(f) ⊆ I + Pα. So rI + Pα. Since R is a graded coprimely packed ring, we have IαΔPα. Then f(I)f(αΔPα). Indeed, if f(I)f(αΔPα), then we have IαΔPα since ker(f ⊆ 1, this gives us a contradiction. Thus we get J=f(I)αΔf(Pα)=αΔPα. Hence S is a graded coprimely packed ring.

Proposition 6. Let R be a graded u –ring. If R is a graded semilocal ring, then R is a graded coprimely packed ring.

Proof. Suppose that R is a graded semilocal ring and hMax(R) = {𝔐1, ..., 𝔐k}. Let I be a graded ideal of R, {Pα}α ∈ ∆ is a family of graded prime ideals of R such that for α ∈ ∆, I + Pα = R. Then there exists a subset {i1, …, it} of {1, … k} for all α ∈ ∆ there exists ij ∈ {i1, …, it} such that Pα ⊆ 𝔐ij. Therefore we get I + 𝔐il= R for all j = 1, …, t. Assume that Itj=1𝔐ij. Since R is a graded u –ring, we get I ⊆ 𝔐il for some {il ∈ {i1, …, it}. And so I + 𝔐il/= R is a contradiction. Thus Itj=1𝔐ij and so IαΔPα.

Proposition 7. Every graded compactly packed ring is a graded coprimely packed ring.

Proof. Suppose that R is a graded compactly packed ring. Let I be graded ideal and {Pα}α ∈ ∆ be a family of graded prime ideals of R such that I + Pα = R for every α ∈ ∆. Assume that IαΔPα. Since R is graded compactly packed ring, we get IPβ for some β ∈ ∆. And so I + Pβ = Pβ/ = R, which is a contradiction. Thus IαΔPα.

Theorem 8. Let R be a graded integral domain and h –dim R = 1. Then R is a graded compactly packed ring if and only if R is a graded coprimely packed ring.

Proof. It is clear that every graded compactly packed ring is a graded coprimely packed ring by Proposition 7. Now suppose that R is a graded coprimely packed ring and IαΔPα, Pα/ = 0 for α ∈ ∆. Assume that I + Pβ/ = R for some β ∈ ∆. Then there exists a graded maximal ideal 𝔐 such that I + Pβ = 𝔐. Since h – dim R = 1, we get Pβ = 𝔐 and so IPβ.

Theorem 9. Let R be a graded ring. Then R is a graded coprimely packed ring if and only if R is coprimely packed ring by hMax(R).

Proof. Suppose that R is a graded coprimely packed ring. Since hMax(R) ⊆ hSpec(R), it is clear that R is coprimely packed ring by hMax(R). Now assume that R is a coprimely packed ring by hMax(R. Let I be graded ideal and Pαα ∈ ∆ be a family of graded prime ideals of R such that I + Pα = R for every α ∈ ∆. Then there exist 𝔐αhMax(R) such that Pα ⊆ 𝔐α. Since I + 𝔐α for every α ∈ ∆, then by our assumption we get IαΔ𝔐α. Hence IαΔPα.

References

  • [1]

    Park C.H., Park M., Integral Closure of a Graded Noetherian Domain, J. Korean Math Soc., 2011, 48, 449-464.

    • Crossref
    • Export Citation
  • [2]

    Farzalipour F., Ghiasvand P., On the Union of Graded Prime Submodules, J. Math., 2011, 9, 49-55.

  • [3]

    Oral K.H., Tekir U., Agargun A.G., On graded prime and primary submodules, Turk. J. Math., 2011, 35, 159-167.

  • [4]

    Quartararo P., Butts H.S., Finite Union of Ideals and Modules. Proceedings of the American Mathematical Society, Proc. Amer. Math. Soc., 1975, 52, 91-96.

    • Crossref
    • Export Citation
  • [5]

    Pakala J.V., Shores T.S., On Compactly Packed Rings, Pacific, J. Math., 1981, 97, 197-201.

    • Crossref
    • Export Citation
  • [6]

    Reis C., Viswanathan T., A compactness property of prime ideals in Noetherian rings, Proc. Amer. Soc., 1970, 25, 353-356.

    • Crossref
    • Export Citation
  • [7]

    Smith W., A covering condition for prime ideals, Proc. Amer. Math. Soc., 1971, 30, 451-452.

    • Crossref
    • Export Citation
  • [8]

    Erdogdu V., Coprimely Packed Rings, J. Number Theor., 1988, 28, 1-5.

    • Crossref
    • Export Citation
  • [9]

    Cho Y.H., Coprimely Packed Rings II, Honam Math. J., 199, 21, 43-47.

  • [10]

    Tekir U., On Coprimely Packed Rings, Commun. Algebra, 2007, 35, 2357-2360.

    • Crossref
    • Export Citation
  • [11]

    Ozkirisci N.A., Oral K.H., Tekir U., Graded prime spectrum of a graded module, IJST, 2013, 37A3, 411-420.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Park C.H., Park M., Integral Closure of a Graded Noetherian Domain, J. Korean Math Soc., 2011, 48, 449-464.

    • Crossref
    • Export Citation
  • [2]

    Farzalipour F., Ghiasvand P., On the Union of Graded Prime Submodules, J. Math., 2011, 9, 49-55.

  • [3]

    Oral K.H., Tekir U., Agargun A.G., On graded prime and primary submodules, Turk. J. Math., 2011, 35, 159-167.

  • [4]

    Quartararo P., Butts H.S., Finite Union of Ideals and Modules. Proceedings of the American Mathematical Society, Proc. Amer. Math. Soc., 1975, 52, 91-96.

    • Crossref
    • Export Citation
  • [5]

    Pakala J.V., Shores T.S., On Compactly Packed Rings, Pacific, J. Math., 1981, 97, 197-201.

    • Crossref
    • Export Citation
  • [6]

    Reis C., Viswanathan T., A compactness property of prime ideals in Noetherian rings, Proc. Amer. Soc., 1970, 25, 353-356.

    • Crossref
    • Export Citation
  • [7]

    Smith W., A covering condition for prime ideals, Proc. Amer. Math. Soc., 1971, 30, 451-452.

    • Crossref
    • Export Citation
  • [8]

    Erdogdu V., Coprimely Packed Rings, J. Number Theor., 1988, 28, 1-5.

    • Crossref
    • Export Citation
  • [9]

    Cho Y.H., Coprimely Packed Rings II, Honam Math. J., 199, 21, 43-47.

  • [10]

    Tekir U., On Coprimely Packed Rings, Commun. Algebra, 2007, 35, 2357-2360.

    • Crossref
    • Export Citation
  • [11]

    Ozkirisci N.A., Oral K.H., Tekir U., Graded prime spectrum of a graded module, IJST, 2013, 37A3, 411-420.

OPEN ACCESS

Journal + Issues

Search