# Two New Types of Rings Constructed from Quasiprime Ideals

Authors: Manal Ghanem; Hassan Al-Ezeh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

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 rings (PF-rings) in commutative rings, especially, for rings of positive characteristic.


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## Body

## 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 ringR is regular if for every element a∈R there exists an element b∈R such that a=a2b. Also a ring R is called a PF-ring if every principal ideal aR of R is an R-flat module. These two types of rings were investigated extensively in the literature, see von Neumann [2], Endo [3], Matlis [4], Goodearl [5], 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 ideal I in a ring R is called a pure ideal if, for each a∈I, there exists b∈I such that ab=a. These ideals classify certain important types of rings, see, for example, Borceux and Van Den Bosch [7] and AL-Ezeh [8, 9].Theorem 1.1 (Goodearl, [5]).
LetR be a ring. Then the following are equivalent.(1)
R is von Neumann regular.(2)
R is reduced and every prime ideal is a maximal one.(3)
Every maximal ideal ofR is pure.(4)
Every element ofR can be written as a product of a unit and an idempotent element.(5)
Every localizationRM at each maximal ideal M is a field.Theorem 1.2 (Goodearl, [5]).
(1) IfR is a von Neumann regular ring and S is a multiplicative subset of R, then ring of fractions S-1R is a von Neumann regular ring.
(2) A direct product of von Neumann regular rings is von Neumann regular.
(3) IfR is von Neumann regular ring and I is an ideal of R, then R/I is von Neumann regular.Theorem 1.3.
LetR be a ring. Then the following are equivalent.(1)
R is PF-ring.(2)
R is reduced and each prime ideal contains a unique minimal prime ideal, see Matlis [4].(3)
For eacha∈R, annR(a)={x∈R:xa=0} is pure ideal in R, see Al-Ezeh [8].(4)
Every localizationRM at each prime ideal M is an integral domain.Theorem 1.4.
IfR is PF-ring and I is pure ideal of R, then R/I is PF-ring.Recall that by aderivation of a ring R we mean any additive map δ:R→R satisfying δ(ab)=δ(a)b+aδ(b) for every a,b∈R. A differential ring R is a ring with a derivation δ. A subset S of R is said to be differential if δ(S)⊆S. For any subset S of R, the set SΔ={x∈S:δ(x)∈S} is called the differential of S. Many properties of SΔ were studied by Keigher in [10, 11]. Let R be a differential ring. Then a differential ideal I is called a quasiprime ideal if there is a multiplicative subset of R such that I is maximal among differential ideals of R disjoint from S. Clearly, a quasiprime ideal of a differential ring is a generalization of a prime ideal of a ring R. Note that I is quasiprime ideal of R if there is a prime ideal P of R such that I=PΔ and r(I)=P, where r(I) is the radical ideal of I in R, see Keigher [10]. 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 in [10–12]. Also, in Keigher [10, 11], the differential rings constructed from quasiprime ideals via quotient rings and rings of fractions were studied too. Recall that a differential ideal I of a differential  ringR is a quasimaximal ideal if r(I) is maximal ideal. So, I is called a quasimaximal ideal if there exists a maximal ideal M such that I=MΔ and r(I)=M. It is clear that every maximal differential ideal is a quasimaximal but the converse need not be true. A differential  ringR is called quasireduced ring if the differential of the nilradical, nil(R), equals zero (i.e., Δ-nil(R)=OR).Now, we define two new types of differential rings that can be constructed using quasiprime ideals.Definition 1.5.
A differential ring(R,δ) is said to be quasiregular if R is quasireduced and every quasiprime ideal is quasimaximal.Definition 1.6.
A differential ring(R,δ) is called a quasi-PF ring if R is quasireduced and every quasiprime ideal of it contains a unique minimal quasiprime ideal.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.
LetS be a multiplicative subset with nonzero divisors of R, then R is quasireduced if and only if S-1R is quasireduced.The following theorem was proved by Keigher in [10].Theorem 2.2.
LetR be a differential ring, and let S be a multiplicative subset of R.(1)
IfP is a prime ideal such that P∩S=ϕ, then in the differential ring S-1R, we have [S-1P]Δ=S-1PΔ.(2)
There is a one to one correspondence between quasiprime ideals inS-1R and quasiprime ideals in R disjoint from S.Now, we can conclude the following.Theorem 2.3.
IfR is a quasiregular ring and S is a multiplicative subset, which does not contain any zero divisors of R, then S-1R is a quasiregular ring.Proof.
By Lemma2.1, if every quasiprime ideal of S-1R is quasimaximal, then S-1R is quasiregular ring. Let P be a prime ideal of S-1R such that r(PΔ)=P. Then there exists a prime ideal T of R disjoint from S such that r(TΔ)=T, S-1TΔ=PΔ and S-1T=P. Since R is a quasiregular ring and TΔ is a quasiprime ideal of R, we see that T is a maximal ideal of R and hence P is a maximal ideal of S-1R.It is well known that ifR is a reduced ring, and let I is an ideal of R, then the factor ring R/I is reduced if and only if r(I)=I. Also one can easily show that if I is a pure ideal of a reduced ring R, then r(I)=I and hence R/I is reduced ring. We can generalize these results to differential rings as follows.Theorem 2.4.
LetR be a quasireduced ring and I be a differential ideal of R.(1)
The factor ringR/I is quasireduced if and only if [r(I)]Δ=I.(2)
I is a pure ideal of R implies that [r(I)]Δ=I.(3)
I is pure ideal of R implies that the factor ring R/I is quasireduced.Proof.
(1) Obvious.
(2) First note that every pure ideal of a differential ring is a differential ideal. Now, suppose thatR is quasireduced ring and I is pure ideal of R. Let x∈[r(I)]Δ. Then there exist positive integers m, n such that xn=xnt and (δ(x))m=(δ(x))mr for some t,r∈I. Since t∈I and I is a differential ideal, there exists k∈I such that δ(t)=δ(t)k. Take y=x(1-t)(1-k)(1-r). Then it is easy to verify that y∈Δ-nil(R). But R is a quasireduced ring, so y=0. Therefore, x=xl, where l=r+t-tr+k[1-r-t+tr]∈I. Thus x∈I.
(3) It follows easily from (1) and (2).Now, we can determine when a factor ringR/I of a quasiregular ring R is quasiregular.Theorem 2.5.
LetR be a quasiregular ring and I be a differential ideal of R. Then the factor ring R/I is a quasiregular ring if and only if [r(I)]Δ=I.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 [r(PΔ)]=P. Then there exists a prime ideal T of R such that [r(TΔ)]=T and P=T/I, because for any differential ideal I of R, there is one to one correspondence between quasiprime ideals in R/I and quasiprime ideals in R that contain I, see Proposition  1.12 in [10]. Since R is a quasiregular ring, T is a maximal ideal of R. Therefore, P is a maximal ideal of R/I.The following corollary follows directly from Theorems2.4 and 2.5.Corollary 2.6.
LetR be a quasiregular ring. For any pure ideal I of R, the ring R/I is 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.
LetR1 and R2 be two differential rings, and let R=R1×R2 and πi:R→Ri, i=1,2, be the two projections. If I is an ideal of R, then(1)
πi(r(I))=r(πi(I)), i=1,2,(2)
πi(IΔ)=[πi(I)]Δ, i=1,2, and(3)
r(IΔ)=I implies that πi(I)=r([πi(I)]Δ), i=1,2.Proof.
(1)x∈π1(r(I)) if and only if (x,0)∈r(I) if and only if (xn,0)∈I if and only if xn∈π1(I) if and only if x∈r(π1(I)).
(2)x∈π1(IΔ) if and only if (x,0)∈I and (δ(x),0)∈I if and only if x∈π1(I) and δ(x)∈π1(I) if and only if x∈[π1(I)]Δ.
(3) It follows easily from (1) and (2).Theorem 2.8.
LetR=∏i=1nRi where Ri is a quasiregular ring. Then R is a quasiregular ring.Proof.
We give the proof for the product of two quasiregular ringsR1 and R2. The general result follows by induction. Let R=R1×R2 where R1 and R2 are quasiregular rings. Since nil(R)=nil(R1)×nil(R2), we have [Δ-nil(R)]=[Δ-nil(R1)]×[Δ-nil(R2)]. But Ri, i=1,2, are quasiregular rings so, Δ-nil(Ri)=ORi, i=1,2. Therefore, Δ-nil(R)=OR and hence R is quasireduced. We may assume that P=P1×R2 is a prime ideal of R such that r(PΔ)=P. From Lemma 2.7, we conclude that r((P1)Δ)=P1. So, (P1)Δ is a quasiprime ideal of R1 and hence it is a quasimaximal ideal of R1. Thus P is a maximal ideal of R.Keigher in [10] introduced the following definitions of differential rings.(1)
R is said to be a q-local ring if R is a local ring whose unique maximal ideal M satisfies r(MΔ)=M (i.e., MΔ is quasimaximal).(2)
R is a quasidomain ring if R is quasireduced and every zero divisor in R is nilpotent.(3)
R is called a quasifield if R is quasireduced and every nonunit of R is nilpotent.It is clear thatR is a quasidomain if and only if OR is a quasiprime ideal. Also, R is a quasifield if and only if OR is a quasimaximal ideal. So, every quasidomain is a quasifield, and every quasifield is q-local. For more details about these classes of rings see Keigher [10].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 2.9 (Keigher, [10]).
Suppose thatR has characteristic l>0, and let P be a prime ideal in R. Then r(PΔ)=P.So, one can conclude the following.Corollary 2.10.
LetR be a differential ring of positive characteristic. Then there is a one to one correspondence between quasiprime ideals in R and prime ideals in R.Now, we give the following result which is analogous to the corresponding one in commutative rings.Theorem 2.11.
LetR be a differential ring of positive characteristic. Then a q-local ring R is quasiregular if and only if R is a quasifield.Proof.
SinceR is a q-local ring, nil(R) is a prime ideal. From Theorem 2.11, we get r(Δ-nil(R))=nil(R). Since R is a quasiregular ring, Δ-nil(R)=OR and nil(R) is a maximal ideal of R. Thus R is a quasifield.
Conversely, it is clear that in any,R being a quasifield implies that R is quasiregular.For rings of positive characteristic, as in von Neumann regular rings we can characterize quasiregular ring by localizations.Theorem 2.12.
LetR be a differential ring of positive characteristic. Then, R is quasiregular if RM is a quasifield for each maximal ideal M of R.Proof.
LetP be a prime ideal of R such that r(PΔ)=P. Then there exists a maximal ideal M of R such that P⊆M and r(MΔ)=M. Therefore, PRM is a prime ideal of RM and r(PΔRM)=PRM. Since RM is a quasifield, we have PΔRM=ORRM=MΔRM. By Corollary 2.10, PΔ=MΔ=OR. Hence P=M and Δ-nil(R)=OR.For a ringR, denote by Z(R) and J(R) the set of zero divisors and Jacobson radical, respectively.As a simple consequence of Theorems2.3, 2.11, and 2.12 we have the following.Theorem 2.13.
LetR be a ring of positive characteristic with Z(R)⊆J(R). Then, R is quasiregular if and only if every localization RM at each maximal ideal M of R is quasifield.


## 3. Quasi-PF Ring

Recall that a ringR is a PF-ring if and only if it is reduced and every prime ideal of R contains a unique minimal prime ideal. So, one can introduce quasi-PF rings. A ring R is called a quasi-PF ring if R is quasireduced and every quasiprime ideal of R 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 ideal I of a quasi-PF ring R, the factor ring R/I is a quasi-PF ring. This follows directly from Theorem  1.12 of [10] and Theorem 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 of R, then S-1R is 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 thatR is a differential ring with positive characteristic. A localization RP of a quasi-PF ring R is a quasidomain for each prime ideal P of R.Proof.
LetP be a prime ideal of R. Then r(PΔ)=P and RP has unique maximal ideal, PRP. Since R is quasi-PF ring with positive characteristic, PRP has unique minimal prime ideal TRP where T is a unique minimal prime ideal of P in R such that r(TΔ)=T. Consequently, nil(RP)=TRP. Furthermore, RP is a quasireduced ring, hence ORP is a quasiprime ideal of RP.Theorem 3.2.
A differential ringR is a quasi-PF ring if every localization RP is a quasidomain for each prime ideal P of R with r(PΔ)=P.Proof.
LetP be a prime ideal of R with r(PΔ)=P. Then RP is quasidomain and hence ORRP is a unique quasiminimal prime ideal of RP. Let T and S be two minimal prime ideals of P with r(TΔ)=T and r(SΔ)=S. Then TΔRP=SΔRP=ORRP. But, there is a one-to-one correspondence between quasiprime ideals of RP and quasiprime ideals of R contained in P. So TΔ=SΔ=OR. Consequentially, T=S and Δ-nil(R)=OR.Now, we prove an analogous result for localizations of maximal ideals.Theorem 3.3.
LetR be a differential ring with positive characteristic and Z(R)⊆J(R). Then, R is a quasi-PF ring if and only if RM is a quasidomain for every maximal ideal M of R.Proof.
(⇒) Let M be a maximal ideal of R. Since R is a quasireduced ring with positive characteristic and Z(R)⊆J(R), to prove that RM is a quasidomain it is enough to show that nil(RM) is a prime ideal of RM. But R is a quasi-PF ring so, the maximal ideal M has a unique minimal prime ideal P of R. Hence PRM is a unique minimal prime ideal of RM. Thus, nil(RM)=PRM.
(⇐) It follows directly from Theorem 3.1.


## 4. Hurwitz Series

The Hurwitz series ring overR is denoted by HR and is defined as follows. The elements of HR are functions a:ℕ→R, where ℕ is the set of natural numbers and a is a sequence of the form(an). The operation of addition in HR is componentwise and for each a=(an), b=(bn)∈HR multiplication is defined by (an)(bn)=(cn), where cn=∑k=0nCknakbn-k for all n∈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 Marttin [14], Fliess [15], and Taft [16]. 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 in [11] and Keigher and Pritchard in [17] demonstrated that the ring HR of Hurwitz series over a commutative ring R with unity is very important in differential algebra. Some properties, which are shared between R and HR have been studied by Keigher [11], Liu [18]. The structure of Hurwitz series of positive characteristic is very close to the structure of R. Accordingly, for ring of positive characteristic, we prove that R is regular (resp., quasi-PF) if 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→R defined as follows: for any (an)∈HR, εR((an))=(an+1) is a derivative of HR, a shift operator, making (HR,δR) a differential ring. For any ideal I of R, Keigher in [10], defined a differential ideal HI of HR as follows: HI={(an)∈HR:an∈I,∀n∈ℕ} and he proved that HR/HI≅H(R/I).Theorem 4.1 (Keigher, [11]).
LetR be a ring with positive characteristic l.(1)
ch(HR)=l.(2)
For anyx=(xn)∈HR, xl=(x0l,0,0,0,…).(3)
IfI is an ideal of R then r(HI)=εR-1(r(I)).(4)
HR is quasireduced if and only if R is reduced.Now, we prove the following new theorem which is the key to our main results of this section.Theorem 4.2.
Letch(R)=l>0. (1)
IfP is prime ideal of HR, then εR(P) is a prime ideal of R.(2)
P is a prime ideal of HR if and only if P=εR-1(T) for some prime ideal T of R.(3)
There is a one-to-one correspondence between prime ideals inR and quasiprime ideals in HR.Proof.
(1) LetP be a prime ideal of HR. Let xy∈εR(P). Then P has an element t with 0th term xy. Therefore, tl=((xy)l,0,0,…)=(xl,0,0,0,…)(yl,0,0,0,…)∈P. Hence (x,0,0,0,…) or (y,0,0,0,…) belongs to P and thus x∈εR(P) or y∈εR(P).
(2) Suppose thatP is a prime ideal of HR. Then εR(P) is a prime ideal of R. Let r∈εR-1(εR(P)) and r0 be the 0th term of r. Then r0∈εR(P) and hence rl=(r0l,0,0,0,…)∈P. Since P is a prime ideal, r∈P. Now, let T=εR(P). Then P=εR-1(T) where T is a prime ideal of R.
Conversely, note thatεR:HR→R is an epimorphism. So for any prime ideal T of R, εR-1(T) is a prime ideal of HR.
(3) From (1) and (2) the result holds.From the above theorem we get the following result, which was proved differently in Keigher [11].Corollary 4.3.
Letch(R)=l>0. (1)
I is a prime ideal of R if and only if HI is a quasiprime ideal of HR.(2)
I is a maximal ideal of R if and only if HI is a quasimaximal ideal of HR.Proof.
(1)(⇒) Obvious.
(⇐) Suppose that r(HI) is a prime ideal of HR. Then r(HI)=εR-1(T) for some prime ideal T of R. But r(HI)=εR-1(I) and ε is an epimorphism. So, T=I.Now, we prove the following theorem that characterizes when the differential ringHR is quasiregular.Theorem 4.4.
LetR be a ring with ch(R)=l>0. Then R is a regular ring if and only if HR is a quasiregular ring.Proof.
By using Corollary2.10 and Theorem 4.1, it is enough to prove that every prime ideal of R is a maximal ideal if and only if every prime ideal of HR is maximal.
LetT be a prime ideal of HR. Then T=εR-1(P) for some prime ideal P of R. Hence P is a maximal ideal of R. Thus T is a maximal ideal of HR.
Conversely, letP be a prime ideal of R. Then HP is a quasiprime ideal of HR. Therefore, HP is a quasimaximal ideal of HR and thus P is a maximal ideal of R.Now, we give a similar result for whenHR is a quasi-PF ring.Theorem 4.5.
LetR be a ring with ch(R)=l>0. Then R is a PF-ring if and only if HR is a quasi-PF ring.Proof.
Note thatT is prime ideal of HR if and only if T=εR-1(P), P is prime ideal of R. Moreover, T0 is a unique minimal prime ideal of T if and only if T0=εR-1(P0), P0 is a unique minimal prime ideal contains in P.From Theorems4.4 and 4.5 we can obtain the following.Theorem 4.6.
LetR be a ring with ch(R)=l>0. (1)
IfR is a regular ring and I is an ideal of R, then HR/HI is a quasiregular ring.(2)
IfR is a PF-ring and I is a pure ideal of R, then HR/HI is a a quasi-PF ring.Proof.
Note that,HR/HI≅H(R/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,Hℤ2[x] is quasi-PF ring but not quasiregular ring since ℤ2[x] is PF-ring but not regular ring. Hℤ6 is a quasi-PF ring but not a quasidomain because ℤ6 is a PF-ring but not an integral domain, see Theorem  4.3 of Keigher [11].Open Questions.
(1) Give alternative characterizations of quasiregular rings and quasi-PF rings.
(2) Is it true thatR is a quasiregular rings if and only if every differential ideal I is pure?



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