Two New Types of Rings Constructed from Quasiprime Ideals

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Volume 2011, Article ID 473413,9pages doi:10.1155/2011/473413

Research Article

Two New Types of Rings Constructed from Quasiprime Ideals

Manal Ghanem

¹

and Hassan Al-Ezeh

²

1Department of Mathematics, Irbid National University, Irbid 21110, Jordan

2Department of Mathematics, Jordan University, Amman 11942, Jordan

Correspondence should be addressed to Manal Ghanem,dr [email protected] Received 23 October 2010; Accepted 16 March 2011

Academic Editor: Jianming Zhan

Copyrightq2011 M. Ghanem and H. Al-Ezeh. This is an open access 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.

Keigher showed that quasi-prime ideals in differential commutative rings are analogues of prime ideals in commutative rings. In that direction, he introduced and studied new types of differential rings using quasi-prime ideals of a differential ring. In the same sprit, we define and study two new types of differential rings which lead to the mirrors of the corresponding results on von Neumann regular rings and principally flat ringsPF-rings in commutative rings, especially, for rings of positive characteristic.

1. Introduction

The derivatives of rings play important roles in ring theory. In particular, they are used to define various ring constructions, for example, see Sections 3.4 to 3.7 of the monograph 1.

Rings considered in this paper are all commutative with unity. Recall that a ringRis regular if for every element a ∈ R there exists an elementb ∈ R such that a a²b. Also a ringRis called a PF-ring if every principal idealaRofRis anR-flat module. These two types of rings were investigated extensively in the literature, see von Neumann2, Endo3, Matlis4, Goodearl5, and Abu-Osba et al.6. In this paper, we generalize these concepts to ordinary diﬀerential rings. Some well-known properties of regular rings and PF-rings are given in the following theorems. Before that, recall that an idealIin a ringRis called a pure ideal if, for eacha ∈ I, there exists b ∈ I such thatab a. These ideals classify certain important types of rings, see, for example, Borceux and Van Den Bosch7and AL-Ezeh 8,9.

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Theorem 1.1Goodearl,5. LetRbe a ring. Then the following are equivalent.

1Ris von Neumann regular.

2Ris reduced and every prime ideal is a maximal one.

3Every maximal ideal ofRis pure.

4Every element ofRcan be written as a product of a unit and an idempotent element.

5Every localizationRMat each maximal idealMis a field.

Theorem 1.2Goodearl, 5. 1IfR is a von Neumann regular ring andS is a multiplicative subset ofR, then ring of fractionsS⁻¹Ris a von Neumann regular ring.

2A direct product of von Neumann regular rings is von Neumann regular.

3IfRis von Neumann regular ring andIis an ideal ofR, thenR/Iis von Neumann regular.

Theorem 1.3. LetRbe a ring. Then the following are equivalent.

1Ris PF-ring.

2Ris reduced and each prime ideal contains a unique minimal prime ideal, see Matlis [4].

3For eacha∈R, annRa {x∈R:xa0}is pure ideal inR, see Al-Ezeh [8].

4Every localizationRMat each prime idealMis an integral domain.

Theorem 1.4. IfRis PF-ring andIis pure ideal ofR, thenR/Iis PF-ring.

Recall that by a derivation of a ringRwe mean any additive mapδ:R → Rsatisfying δab δabaδbfor everya, b ∈ R. A diﬀerential ringRis a ring with a derivationδ.

A subsetSofRis said to be differential ifδS ⊆S. For any subsetSofR, the setS_Δ {x∈ S : δx ∈ S}is called the differential ofS. Many properties ofSΔ were studied by Keigher in10,11. LetRbe a differential ring. Then a differential idealI is called a quasiprime ideal if there is a multiplicative subset ofR such thatI is maximal among differential ideals ofR disjoint fromS. Clearly, a quasiprime ideal of a differential ring is a generalization of a prime ideal of a ringR. Note thatIis quasiprime ideal ofRif there is a prime idealPofRsuch that I P_ΔandrI P, whererIis the radical ideal ofIinR, see Keigher10. Every prime differential ideal is quasiprime, while the converse need not be true. Also every maximal differential ideal is quasiprime but it need not be prime differential ideal. Quasiprime ideals were studied extensively by Keigher in10–12. Also, in Keigher10,11, the differential rings constructed from quasiprime ideals via quotient rings and rings of fractions were studied too.

Recall that a differential idealIof a differential ringRis a quasimaximal ideal ifrIis maximal ideal. So,Iis called a quasimaximal ideal if there exists a maximal idealMsuch thatIM_Δ and rI M. It is clear that every maximal differential ideal is a quasimaximal but the converse need not be true. A differential ringRis called quasireduced ring if the differential of the nilradical, nilR, equals zeroi.e.,Δ−nilR O_R.

Now, we define two new types of diﬀerential rings that can be constructed using quasiprime ideals.

Definition 1.5. A diﬀerential ringR, δ is said to be quasiregular ifR is quasireduced and every quasiprime ideal is quasimaximal.

Definition 1.6. A diﬀerential ringR, δis called a quasi-PF ring ifRis quasireduced and every quasiprime ideal of it contains a unique minimal quasiprime ideal.

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It is clear that the concept of quasiregular rings and quasi-PF rings are generalizations to the diﬀerential context of von Neumann regular rings and PF-rings in ordinary commutative rings. Our aim in this paper is to study the classes of quasiregular rings and quasi-PF rings and how their structures closely mirrors that of classes of von Neumann regular rings and PF-rings in commutative rings. Also, we investigate when the Hurwitz series ring is quasiregular or quasi-PF, in particular, for rings of positive characteristic.

2. Quasiregular Ring

In this section we study some basic properties of quasiregular rings. We will show that the structure of these classes of rings is very closely connected to the structure of the corresponding class in commutative rings, especially, for rings of positive characteristic. We start by stating an easy lemma that will be used frequently later on.

Lemma 2.1. LetSbe a multiplicative subset with nonzero divisors ofR, thenRis quasireduced if and only ifS⁻¹Ris quasireduced.

The following theorem was proved by Keigher in10.

Theorem 2.2. LetRbe a diﬀerential ring, and letSbe a multiplicative subset ofR.

1If P is a prime ideal such that P ∩S φ, then in the diﬀerential ringS⁻¹R, we have S⁻¹P_ΔS⁻¹PΔ.

2There is a one to one correspondence between quasiprime ideals inS⁻¹Rand quasiprime ideals inRdisjoint fromS.

Now, we can conclude the following.

Theorem 2.3. IfRis a quasiregular ring andSis a multiplicative subset, which does not contain any zero divisors ofR, thenS⁻¹Ris a quasiregular ring.

Proof. By Lemma 2.1, if every quasiprime ideal of S⁻¹R is quasimaximal, then S⁻¹R is quasiregular ring. LetP be a prime ideal ofS⁻¹Rsuch thatrP_Δ P. Then there exists a prime idealTofRdisjoint fromSsuch thatrTΔ T,S⁻¹TΔPΔandS⁻¹T P. SinceRis a quasiregular ring andTΔis a quasiprime ideal ofR, we see thatTis a maximal ideal ofRand hencePis a maximal ideal ofS⁻¹R.

It is well known that ifRis a reduced ring, and letIis an ideal ofR, then the factor ringR/Iis reduced if and only ifrI I. Also one can easily show that ifIis a pure ideal of a reduced ringR, thenrI I and henceR/Iis reduced ring. We can generalize these results to diﬀerential rings as follows.

Theorem 2.4. LetRbe a quasireduced ring andIbe a diﬀerential ideal ofR.

1The factor ringR/Iis quasireduced if and only ifrI_ΔI.

2Iis a pure ideal ofRimplies thatrI_ΔI.

3Iis pure ideal ofRimplies that the factor ringR/Iis quasireduced.

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Proof. 1Obvious.

2First note that every pure ideal of a diﬀerential ring is a diﬀerential ideal. Now, suppose that R is quasireduced ring andI is pure ideal of R. Let x ∈ rI_Δ. Then there exist positive integersm,n such thatxⁿ xⁿtand δx^m δx^mr for some t, r ∈ I.

Sincet ∈ I andI is a diﬀerential ideal, there existsk ∈ I such thatδt δtk. Takey x1−t1−k1−r. Then it is easy to verify thaty∈Δ−nilR. ButRis a quasireduced ring, soy0. Therefore,xxl, wherelrt−trk1−r−ttr∈I. Thusx∈I.

3It follows easily from1and2.

Now, we can determine when a factor ringR/Iof a quasiregular ringRis quasiregular.

Theorem 2.5. LetRbe a quasiregular ring andIbe a diﬀerential ideal ofR. Then the factor ringR/I is a quasiregular ring if and only ifrI_ΔI.

Proof. ⇒Obvious.

⇐ It is enough to show that every quasiprime ideal of R/I is quasimaximal. Let P be a prime ideal of R/I such that rPΔ P. Then there exists a prime ideal T of R such thatrT_Δ T and P T/I, because for any diﬀerential idealI ofR, there is one to one correspondence between quasiprime ideals inR/I and quasiprime ideals in R that containI, see Proposition 1.12 in10. SinceRis a quasiregular ring,T is a maximal ideal of R. Therefore,Pis a maximal ideal ofR/I.

The following corollary follows directly from Theorems2.4and2.5.

Corollary 2.6. LetRbe a quasiregular ring. For any pure idealIofR, the ringR/Iis quasiregular.

Next, we will show that a finite direct product of quasiregular rings is quasiregular.

The following observation is trivial but useful for our purpose.

Lemma 2.7. LetR1andR2be two diﬀerential rings, and letRR1×R2andπi:R → Ri,i1,2, be the two projections. IfIis an ideal ofR, then

1π_irI rπ_iI,i1,2, 2πiIΔ πiI_Δ,i1,2, and

3rIΔ Iimplies thatπiI rπiI_Δ,i1,2.

Proof. 1 x ∈ π1rIif and only if x,0 ∈ rI if and only ifxⁿ,0 ∈ I if and only if xⁿ∈π1Iif and only ifx∈rπ1I.

2x ∈ π1IΔif and only ifx,0 ∈I andδx,0 ∈ I if and only ifx ∈ π1Iand δx∈π1Iif and only ifx∈π1I_Δ.

3It follows easily from1and2.

Theorem 2.8. LetR_n

i1RiwhereRiis a quasiregular ring. ThenRis a quasiregular ring.

Proof. We give the proof for the product of two quasiregular rings R1 and R2. The general result follows by induction. LetR R1×R2 whereR1 andR2 are quasiregular rings. Since nilR nilR1×nilR2, we haveΔ−nilR Δ−nilR1×Δ−nilR2. ButRi,i1,2, are quasiregular rings so,Δ−nilR_i O_R_i,i1,2. Therefore,Δ−nilR O_Rand henceR

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is quasireduced. We may assume thatP P1×R2 is a prime ideal ofRsuch thatrPΔ P.

FromLemma 2.7, we conclude thatrP1_Δ P1. So,P1_Δis a quasiprime ideal ofR1and hence it is a quasimaximal ideal ofR1. ThusPis a maximal ideal ofR.

Keigher in10introduced the following definitions of diﬀerential rings.

1R is said to be a q-local ring if Ris a local ring whose unique maximal ideal M satisfiesrM_Δ Mi.e.,M_Δis quasimaximal.

2Ris a quasidomain ring ifRis quasireduced and every zero divisor inRis nilpotent.

3Ris called a quasifield ifRis quasireduced and every nonunit ofRis nilpotent.

It is clear thatRis a quasidomain if and only ifO_Ris a quasiprime ideal. Also,Ris a quasifield if and only ifOR is a quasimaximal ideal. So, every quasidomain is a quasifield, and every quasifield isq-local. For more details about these classes of rings see Keigher10.

Next, we give a characterization ofq-local quasiregular rings and a characterization of quasiregular rings through localization, when the ring is of positive characteristic. First, we state the following result which is quite helpful.

Theorem 2.9Keigher,10. Suppose thatRhas characteristicl >0, and letP be a prime ideal in R. ThenrPΔ P.

So, one can conclude the following.

Corollary 2.10. Let R be a diﬀerential ring of positive characteristic. Then there is a one to one correspondence between quasiprime ideals inRand prime ideals inR.

Now, we give the following result which is analogous to the corresponding one in commutative rings.

Theorem 2.11. Let R be a diﬀerential ring of positive characteristic. Then a q-local ring R is quasiregular if and only ifRis a quasifield.

Proof. Since R is a q-local ring, nilR is a prime ideal. From Theorem 2.11, we get rΔ− nilR nilR. SinceRis a quasiregular ring,Δ−nilR ORand nilRis a maximal ideal ofR. ThusRis a quasifield.

Conversely, it is clear that in any,Rbeing a quasifield implies thatRis quasiregular.

For rings of positive characteristic, as in von Neumann regular rings we can character- ize quasiregular ring by localizations.

Theorem 2.12. LetRbe a diﬀerential ring of positive characteristic. Then,Ris quasiregular ifRMis a quasifield for each maximal idealMofR.

Proof. LetPbe a prime ideal ofRsuch thatrP_Δ P. Then there exists a maximal idealMof Rsuch thatP ⊆MandrMΔ M. Therefore,PRMis a prime ideal ofRMandrPΔRM

PRM. Since RM is a quasifield, we have PΔRM ORRM MΔRM. By Corollary 2.10, P_ΔM_ΔO_R. HenceP MandΔ−nilR O_R.

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For a ringR, denote byZRandJRthe set of zero divisors and Jacobson radical, respectively.

As a simple consequence of Theorems2.3,2.11, and2.12we have the following.

Theorem 2.13. LetRbe a ring of positive characteristic withZR⊆JR. Then,Ris quasiregular if and only if every localizationR_Mat each maximal idealMofRis quasifield.

3. Quasi-PF Ring

Recall that a ringRis a PF-ring if and only if it is reduced and every prime ideal ofRcontains a unique minimal prime ideal. So, one can introduce quasi-PF rings. A ringRis called a quasi- PF ring ifR is quasireduced and every quasiprime ideal ofR contains a unique minimal quasiprime ideal. It is clear that every quasiregular ring is a quasi-PF ring and that every quasidomain is a quasi-PF ring. For a pure idealI of a quasi-PF ringR, the factor ringR/I is a quasi-PF ring. This follows directly from Theorem 1.12 of10 andTheorem 2.4. From Lemma 2.1 and Theorem 2.2, we can conclude that, if R is quasi-PF ring and S is a multiplicative subset with nonzero divisors ofR, thenS⁻¹Ris a quasi-PF ring. Furthermore, for rings with positive characteristic, a localization of quasi-PF ring is a quasidomain. This result is given in the following theorem.

Theorem 3.1. Suppose thatRis a diﬀerential ring with positive characteristic. A localizationR_P of a quasi-PF ringRis a quasidomain for each prime idealPofR.

Proof. LetP be a prime ideal ofR. ThenrPΔ P andRP has unique maximal ideal,PRP. SinceR is quasi-PF ring with positive characteristic,PR_P has unique minimal prime ideal TR_P whereT is a unique minimal prime ideal ofP inRsuch thatrT_Δ T. Consequently, nilRP TRP. Furthermore,RP is a quasireduced ring, henceORP is a quasiprime ideal of R_P.

Theorem 3.2. A diﬀerential ringRis a quasi-PF ring if every localizationR_P is a quasidomain for each prime idealP ofRwithrPΔ P.

Proof. LetP be a prime ideal ofRwithrP_Δ P. ThenR_P is quasidomain and henceO_RR_P is a unique quasiminimal prime ideal ofRP. Let T and Sbe two minimal prime ideals of P withrTΔ T and rSΔ S. ThenTΔRP SΔRP ORRP. But, there is a one-to-one correspondence between quasiprime ideals ofR_P and quasiprime ideals ofRcontained inP.

SoT_ΔS_Δ O_R. Consequentially,T SandΔ−nilR O_R.

Now, we prove an analogous result for localizations of maximal ideals.

Theorem 3.3. LetRbe a diﬀerential ring with positive characteristic andZR⊆JR. Then,Ris a quasi-PF ring if and only ifR_Mis a quasidomain for every maximal idealMofR.

Proof. ⇒ Let M be a maximal ideal of R. Since R is a quasireduced ring with positive characteristic and ZR ⊆ JR, to prove that RM is a quasidomain it is enough to show that nilR_Mis a prime ideal of R_M. ButRis a quasi-PF ring so, the maximal idealMhas a unique minimal prime idealP ofR. HencePRM is a unique minimal prime ideal ofRM. Thus, nilRM PRM.

⇐It follows directly fromTheorem 3.1.

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4. Hurwitz Series

The Hurwitz series ring overRis denoted by HR and is defined as follows. The elements of HR are functionsa: → R, where is the set of natural numbers andais a sequence of the forman. The operation of addition in HR is componentwise and for eacha an, b b_n∈HR multiplication is defined bya_nb_n c_n, wherec_n _n

k0Cⁿ_ka_kb_n−kfor all n ∈ . It can be easily shown that HR is a ring with zero element 0 0,0,0, . . .,0, . . ., the unity of this ring is the sequence with 0th term 1 and nth term 0 for all n ≥ 1. The ring HR has been named the ring of Hurwitz series in honors to Hurwitz who was the first to consider the product of sequences using the binomial coefficients 13. The product of sequences using the binomial coefficients was studied extensively, for example, see Bochner and Marttin14, Fliess15, and Taft16. The ring of Hurwitz series has been of interest and has had important applications in many areas. In the discussion of weak normalization 4. In differential algebra, Keigher in11and Keigher and Pritchard in17demonstrated that the ring HR of Hurwitz series over a commutative ringRwith unity is very important in differential algebra. Some properties, which are shared betweenRand HR have been studied by Keigher11, Liu18. The structure of Hurwitz series of positive characteristic is very close to the structure ofR. Accordingly, for ring of positive characteristic, we prove thatR is regularresp., quasi-PFif and only if HR is quasiregular resp., quasi-PF. But before that, recall from Keigher 11 that for any ring R there is a natural ring homomorphism εR : HR → Rdefined as follows: for any an ∈ HR,εRan an1is a derivative of HR, a shift operator, makingHR, δ_Ra differential ring. For any idealIofR, Keigher in10, defined a differential ideal HI of HR as follows: HI {an∈ HR :an ∈I, ∀n∈ } and he proved that HR/HI∼HR/I.

Theorem 4.1Keigher,11. LetRbe a ring with positive characteristicl.

1chHR l.

2For anyx x_n∈HR,x^l x^l₀,0,0,0, . . ..

3IfIis an ideal ofRthenrHI ε_R⁻¹rI.

4HR is quasireduced if and only ifRis reduced.

Now, we prove the following new theorem which is the key to our main results of this section.

Theorem 4.2. Let chR l >0.

1IfPis prime ideal of HR, thenε_RPis a prime ideal ofR.

2Pis a prime ideal of HR if and only ifP ε⁻¹_R Tfor some prime idealTofR.

3There is a one-to-one correspondence between prime ideals inRand quasiprime ideals in HR.

Proof. 1LetPbe a prime ideal of HR. Letxy∈ε_RP. ThenPhas an elementtwith 0th term xy. Therefore,t^l xy^l,0,0, . . . x^l,0,0,0, . . .y^l,0,0,0, . . . ∈P. Hencex,0,0,0, . . .or y,0,0,0, . . .belongs toP and thusx∈ε_RPory∈ε_RP.

2Suppose thatP is a prime ideal of HR. ThenεRPis a prime ideal of R. Letr ∈ ε⁻¹_RεRPandr0be the 0th term ofr. Thenr0∈εRPand hencer^l r₀^l,0,0,0, . . .∈P. Since Pis a prime ideal,r∈P. Now, letT ε_RP. ThenPε_R⁻¹TwhereTis a prime ideal ofR.

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Conversely, note thatεR: HR → Ris an epimorphism. So for any prime idealTofR, ε⁻¹_RTis a prime ideal of HR.

3From1and2the result holds.

From the above theorem we get the following result, which was proved diﬀerently in Keigher11.

Corollary 4.3. Let chR l >0.

1Iis a prime ideal ofRif and only if HI is a quasiprime ideal of HR.

2Iis a maximal ideal ofRif and only if HI is a quasimaximal ideal of HR.

Proof. 1 ⇒Obvious.

⇐Suppose thatrHIis a prime ideal of HR. ThenrHI ε⁻¹_R Tfor some prime idealT ofR. ButrHI ε⁻¹_R Iandεis an epimorphism. So,T I.

Now, we prove the following theorem that characterizes when the diﬀerential ring HR is quasiregular.

Theorem 4.4. LetRbe a ring with chR l > 0. ThenRis a regular ring if and only if HR is a quasiregular ring.

Proof. By usingCorollary 2.10andTheorem 4.1, it is enough to prove that every prime ideal ofRis a maximal ideal if and only if every prime ideal of HR is maximal.

LetTbe a prime ideal of HR. ThenT ε⁻¹_RPfor some prime idealPofR. HencePis a maximal ideal ofR. ThusTis a maximal ideal of HR.

Conversely, letPbe a prime ideal ofR. Then HP is a quasiprime ideal of HR. Therefore, HP is a quasimaximal ideal of HR and thusPis a maximal ideal ofR.

Now, we give a similar result for when HR is a quasi-PF ring.

Theorem 4.5. LetRbe a ring with chR l >0. ThenRis a PF-ring if and only if HR is a quasi-PF ring.

Proof. Note that T is prime ideal of HR if and only if T ε⁻¹_RP,P is prime ideal of R.

Moreover,T0is a unique minimal prime ideal ofT if and only ifT0 ε⁻¹_R P0,P0is a unique minimal prime ideal contains inP.

From Theorems4.4and4.5we can obtain the following.

Theorem 4.6. LetRbe a ring with chR l >0.

1IfRis a regular ring andIis an ideal ofR, then HR/HI is a quasiregular ring.

2IfRis a PF-ring andIis a pure ideal ofR, then HR/HI is a a quasi-PF ring.

Proof. Note that, HR/HI∼HR/I.

Remark 4.7. Every quasiregular ring is a quasi-PF ring and every quasidomain is a quasi-PF ring. But the converse is not true. For example,H2xis quasi-PF ring but not quasiregular ring since2xis PF-ring but not regular ring.H6is a quasi-PF ring but not a quasidomain because6is a PF-ring but not an integral domain, see Theorem 4.3 of Keigher11.

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Open Questions. 1 Give alternative characterizations of quasiregular rings and quasi-PF rings.

2Is it true thatR is a quasiregular rings if and only if every diﬀerential ideal I is pure?

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