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 h – Spec(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
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, [5–7]. The ring R is compactly packed if for any ideal I of R,
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
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
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 u−ideal,
(iii) If
(iv) If
Proof. (i) ⇒ (ii) is trivial.
(ii) ⇒ (iii) Since
(iii) ⇒ (iv) Suppose that
(iv) ⇒ (i) It follows from,
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 u−ring, then S−1R is a graded u− ring.
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∈∆,
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 : R → S be an epimorphism. Assume that
Now recall the following well known Lemma.
Lemma 1. [3, Lemma 2.1] Let R be a graded ring, a ∈ h(R) and I, J be graded ideals of R. Then aR, I + J and I ⋂ J 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 S ⊆ h(R) be a multiplicatively closed subset. Then the set
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
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,
(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
(iii) ⇒ (i) Suppose that
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 h –htP ≤ 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 r ∈ h(R) such that
Theorem 5. Let R be a graded ring, I be a graded ideal and P be a graded prime ideal such that I ⊆ P. 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 x ∈ P ⋂ h(R), there is a y ∈ h(R)\Pand a nonnegative integer i such that yxi ∈ I.
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| y ∈ h(R) P, i = 0, 1, 2, ...}. Then h(R)\P ⊊ S. Since h(R)\ P is maximal, there exist an element y ∈ h(R)\ P and i nonnegative integer such that yxi ∈ I.
(iii) ⇒ (i) Assume that I ⊂ Q ⊆ P, where Q is graded prime ideal. If there exists x ∈ P\Q where x ∈ h(R), then there exist an element y ∈ h(R)\P such that yxi ∈ I for some i = 0, 1, 2, .... Therefore yxi ∈ Q, y ∉ Q. Thus xi ∈ Q. It is a contradiction.
Recall that a graded ring R is reduced if its nilradical is zero, i.e.
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 x ∈ P ⋂ h(R) there exists some y ∈ h(R)\ P such that xy = 0.
Theorem 6.
let R be a graded ring and h – Min(R) = {Pα}. If
Proof. Without loss of generality, assume that R is a graded reduced ring. Then for P ∈ h – MinR, Rp is a graded field. Let
Corollary 2. Let R be a graded u-ring and h – Min(R) = {Pα}. Then
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, h –Spec(R), is a topology with the closed sets
Definition 3. Let R be a graded ring, Λ an index set and r, sα ∈ h(R)\{0} for all α ∈ Λ. Then we say that R has property (*)
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 ∈ 𝔐1 ⋂ h(R) and b ∈ 𝔐2 ⋂ h(R) such that a + b = 1R. Now let c ∈ (𝔐3 ⋂ h(R))\(𝔐1 ⋃ 𝔐2). Since c = ca + cb, we get Dc ⊆ Dac ⋃ Dbc. Since R satisfies (*) property we get Dc ⊆ Dac or Dc ⊆ Dbc. 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; then
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 : R → S be an epimorphism. Assume that j be a graded ideal of S and
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 h – Max(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
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
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
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 h – Max(R).
Proof. Suppose that R is a graded coprimely packed ring. Since h – Max(R) ⊆ h – Spec(R), it is clear that R is coprimely packed ring by h – Max(R). Now assume that R is a coprimely packed ring by h – Max(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 𝔐α ∈ h – Max(R) such that Pα ⊆ 𝔐α. Since I + 𝔐α for every α ∈ ∆, then by our assumption we get
References
[1] Park C.H., Park M., Integral Closure of a Graded Noetherian Domain, J. Korean Math Soc., 2011, 48, 449-464.10.4134/JKMS.2011.48.3.449Search in Google Scholar
[2] Farzalipour F., Ghiasvand P., On the Union of Graded Prime Submodules, J. Math., 2011, 9, 49-55.10.5402/2011/939687Search in Google Scholar
[3] Oral K.H., Tekir U., Agargun A.G., On graded prime and primary submodules, Turk. J. Math., 2011, 35, 159-167.10.3906/mat-0904-11Search in Google Scholar
[4] Quartararo P., Butts H.S., Finite Union of Ideals and Modules. Proceedings of the American Mathematical Society, Proc. Amer. Math. Soc., 1975, 52, 91-96.10.1090/S0002-9939-1975-0382249-5Search in Google Scholar
[5] Pakala J.V., Shores T.S., On Compactly Packed Rings, Pacific, J. Math., 1981, 97, 197-201.10.2140/pjm.1981.97.197Search in Google Scholar
[6] Reis C., Viswanathan T., A compactness property of prime ideals in Noetherian rings, Proc. Amer. Soc., 1970, 25, 353-356.10.1090/S0002-9939-1970-0254031-6Search in Google Scholar
[7] Smith W., A covering condition for prime ideals, Proc. Amer. Math. Soc., 1971, 30, 451-452.10.1090/S0002-9939-1971-0282963-2Search in Google Scholar
[8] Erdogdu V., Coprimely Packed Rings, J. Number Theor., 1988, 28, 1-5.10.1016/0022-314X(88)90115-1Search in Google Scholar
[9] Cho Y.H., Coprimely Packed Rings II, Honam Math. J., 199, 21, 43-47.Search in Google Scholar
[10] Tekir U., On Coprimely Packed Rings, Commun. Algebra, 2007, 35, 2357-2360.10.1080/00927870701325611Search in Google Scholar
[11] Ozkirisci N.A., Oral K.H., Tekir U., Graded prime spectrum of a graded module, IJST, 2013, 37A3, 411-420.Search in Google Scholar
© 2016 R.N. Uregen et al., published by De Gruyter Open
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.