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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 473413, 9 pages
http://dx.doi.org/10.1155/2011/473413
Research Article

Two New Types of Rings Constructed from Quasiprime Ideals

1Department of Mathematics, Irbid National University, Irbid 21110, Jordan
2Department of Mathematics, Jordan University, Amman 11942, Jordan

Received 23 October 2010; Accepted 16 March 2011

Academic Editor: Jianming Zhan

Copyright © 2011 Manal Ghanem and Hassan 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.

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.

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 ring is regular if for every element there exists an element such that . Also a ring is called a PF-ring if every principal ideal of is an -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 in a ring is called a pure ideal if, for each , there exists such that . 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]). Let be a ring. Then the following are equivalent.(1) is von Neumann regular.(2) is reduced and every prime ideal is a maximal one.(3)Every maximal ideal of is pure.(4)Every element of can be written as a product of a unit and an idempotent element.(5)Every localization at each maximal ideal is a field.

Theorem 1.2 (Goodearl, [5]). (1) If is a von Neumann regular ring and is a multiplicative subset of , then ring of fractions is a von Neumann regular ring.
(2) A direct product of von Neumann regular rings is von Neumann regular.
(3) If is von Neumann regular ring and is an ideal of , then is von Neumann regular.

Theorem 1.3. Let be a ring. Then the following are equivalent.(1) is PF-ring.(2) is reduced and each prime ideal contains a unique minimal prime ideal, see Matlis [4].(3)For each , is pure ideal in , see Al-Ezeh [8].(4)Every localization at each prime ideal is an integral domain.

Theorem 1.4. If is PF-ring and is pure ideal of , then is PF-ring.

Recall that by a derivation of a ring we mean any additive map satisfying for every . A differential ring is a ring with a derivation . A subset of is said to be differential if . For any subset of , the set is called the differential of . Many properties of were studied by Keigher in [10, 11]. Let be a differential ring. Then a differential ideal is called a quasiprime ideal if there is a multiplicative subset of such that is maximal among differential ideals of disjoint from . Clearly, a quasiprime ideal of a differential ring is a generalization of a prime ideal of a ring . Note that is quasiprime ideal of if there is a prime ideal of such that and , where is the radical ideal of in , 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 [1012]. 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 of a differential ring is a quasimaximal ideal if is maximal ideal. So, is called a quasimaximal ideal if there exists a maximal ideal such that and . It is clear that every maximal differential ideal is a quasimaximal but the converse need not be true. A differential ring is called quasireduced ring if the differential of the nilradical, , equals zero (i.e., ).

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

Definition 1.5. A differential ring is said to be quasiregular if is quasireduced and every quasiprime ideal is quasimaximal.

Definition 1.6. A differential ring is called a quasi-PF ring if 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. Let be a multiplicative subset with nonzero divisors of , then is quasireduced if and only if is quasireduced.

The following theorem was proved by Keigher in [10].

Theorem 2.2. Let be a differential ring, and let be a multiplicative subset of .(1)If is a prime ideal such that , then in the differential ring , we have .(2)There is a one to one correspondence between quasiprime ideals in and quasiprime ideals in disjoint from .

Now, we can conclude the following.

Theorem 2.3. If is a quasiregular ring and is a multiplicative subset, which does not contain any zero divisors of , then is a quasiregular ring.

Proof. By Lemma 2.1, if every quasiprime ideal of is quasimaximal, then is quasiregular ring. Let be a prime ideal of such that . Then there exists a prime ideal of disjoint from such that , and . Since is a quasiregular ring and is a quasiprime ideal of , we see that is a maximal ideal of and hence is a maximal ideal of .

It is well known that if is a reduced ring, and let is an ideal of , then the factor ring is reduced if and only if . Also one can easily show that if is a pure ideal of a reduced ring , then and hence is reduced ring. We can generalize these results to differential rings as follows.

Theorem 2.4. Let be a quasireduced ring and be a differential ideal of .(1)The factor ring is quasireduced if and only if .(2) is a pure ideal of implies that .(3) is pure ideal of implies that the factor ring is quasireduced.

Proof. (1) Obvious.
(2) First note that every pure ideal of a differential ring is a differential ideal. Now, suppose that is quasireduced ring and is pure ideal of . Let . Then there exist positive integers , such that and for some . Since and is a differential ideal, there exists such that . Take . Then it is easy to verify that . But is a quasireduced ring, so . Therefore, , where . Thus .
(3) It follows easily from (1) and (2).

Now, we can determine when a factor ring of a quasiregular ring is quasiregular.

Theorem 2.5. Let be a quasiregular ring and be a differential ideal of . Then the factor ring is a quasiregular ring if and only if .

Proof. Obvious.
It is enough to show that every quasiprime ideal of is quasimaximal. Let be a prime ideal of such that . Then there exists a prime ideal of such that and , because for any differential ideal of , there is one to one correspondence between quasiprime ideals in and quasiprime ideals in that contain , see Proposition  1.12 in [10]. Since is a quasiregular ring, is a maximal ideal of . Therefore, is a maximal ideal of .

The following corollary follows directly from Theorems 2.4 and 2.5.

Corollary 2.6. Let be a quasiregular ring. For any pure ideal of , the ring 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. Let and be two differential rings, and let and , , be the two projections. If is an ideal of , then(1) , ,(2) , , and(3) implies that , .

Proof. (1) if and only if if and only if if and only if if and only if .
(2) if and only if and if and only if and if and only if .
(3) It follows easily from (1) and (2).

Theorem 2.8. Let where is a quasiregular ring. Then is a quasiregular ring.

Proof. We give the proof for the product of two quasiregular rings and . The general result follows by induction. Let where and are quasiregular rings. Since , we have . But , , are quasiregular rings so, , . Therefore, and hence is quasireduced. We may assume that is a prime ideal of such that . From Lemma 2.7, we conclude that . So, is a quasiprime ideal of and hence it is a quasimaximal ideal of . Thus is a maximal ideal of .

Keigher in [10] introduced the following definitions of differential rings.(1) is said to be a -local ring if is a local ring whose unique maximal ideal satisfies (i.e., is quasimaximal).(2) is a quasidomain ring if is quasireduced and every zero divisor in is nilpotent.(3) is called a quasifield if is quasireduced and every nonunit of is nilpotent.

It is clear that is a quasidomain if and only if is a quasiprime ideal. Also, is a quasifield if and only if is a quasimaximal ideal. So, every quasidomain is a quasifield, and every quasifield is -local. For more details about these classes of rings see Keigher [10].

Next, we give a characterization of -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 that has characteristic , and let be a prime ideal in . Then .

So, one can conclude the following.

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

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

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

Proof. Since is a -local ring, is a prime ideal. From Theorem 2.11, we get . Since is a quasiregular ring, and is a maximal ideal of . Thus is a quasifield.
Conversely, it is clear that in any, being a quasifield implies that is quasiregular.

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

Theorem 2.12. Let be a differential ring of positive characteristic. Then, is quasiregular if is a quasifield for each maximal ideal of .

Proof. Let be a prime ideal of such that . Then there exists a maximal ideal of such that and . Therefore, is a prime ideal of and . Since is a quasifield, we have . By Corollary 2.10, . Hence and .

For a ring , denote by and the set of zero divisors and Jacobson radical, respectively.

As a simple consequence of Theorems 2.3, 2.11, and 2.12 we have the following.

Theorem 2.13. Let be a ring of positive characteristic with . Then, is quasiregular if and only if every localization at each maximal ideal of is quasifield.

3. Quasi-PF Ring

Recall that a ring is a PF-ring if and only if it is reduced and every prime ideal of contains a unique minimal prime ideal. So, one can introduce quasi-PF rings. A ring is called a quasi-PF ring if is quasireduced and every quasiprime ideal of 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 of a quasi-PF ring , the factor ring 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 is quasi-PF ring and is a multiplicative subset with nonzero divisors of , then 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 that is a differential ring with positive characteristic. A localization of a quasi-PF ring is a quasidomain for each prime ideal of .

Proof. Let be a prime ideal of . Then and has unique maximal ideal, . Since is quasi-PF ring with positive characteristic, has unique minimal prime ideal where is a unique minimal prime ideal of in such that . Consequently, . Furthermore, is a quasireduced ring, hence is a quasiprime ideal of .

Theorem 3.2. A differential ring is a quasi-PF ring if every localization is a quasidomain for each prime ideal of with .

Proof. Let be a prime ideal of with . Then is quasidomain and hence is a unique quasiminimal prime ideal of . Let and be two minimal prime ideals of with and . Then . But, there is a one-to-one correspondence between quasiprime ideals of and quasiprime ideals of contained in . So . Consequentially, and .

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

Theorem 3.3. Let be a differential ring with positive characteristic and . Then, is a quasi-PF ring if and only if is a quasidomain for every maximal ideal of .

Proof. Let be a maximal ideal of . Since is a quasireduced ring with positive characteristic and , to prove that is a quasidomain it is enough to show that is a prime ideal of . But is a quasi-PF ring so, the maximal ideal has a unique minimal prime ideal of . Hence is a unique minimal prime ideal of . Thus, .
It follows directly from Theorem 3.1.

4. Hurwitz Series

The Hurwitz series ring over is denoted by and is defined as follows. The elements of are functions , where is the set of natural numbers and is a sequence of the form . The operation of addition in is componentwise and for each , multiplication is defined by , where for all . It can be easily shown that is a ring with zero element , the unity of this ring is the sequence with 0th term 1 and th term 0 for all . The ring 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 of Hurwitz series over a commutative ring with unity is very important in differential algebra. Some properties, which are shared between and have been studied by Keigher [11], Liu [18]. The structure of Hurwitz series of positive characteristic is very close to the structure of . Accordingly, for ring of positive characteristic, we prove that is regular (resp., quasi-PF) if and only if is quasiregular (resp., quasi-PF). But before that, recall from Keigher [11] that for any ring there is a natural ring homomorphism defined as follows: for any , is a derivative of , a shift operator, making a differential ring. For any ideal of , Keigher in [10], defined a differential ideal of as follows: and he proved that .

Theorem 4.1 (Keigher, [11]). Let be a ring with positive characteristic .(1) .(2)For any , .(3)If is an ideal of then .(4) is quasireduced if and only if is reduced.

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

Theorem 4.2. Let . (1)If is prime ideal of , then is a prime ideal of .(2) is a prime ideal of if and only if for some prime ideal of .(3)There is a one-to-one correspondence between prime ideals in and quasiprime ideals in .

Proof. (1) Let be a prime ideal of . Let . Then has an element with 0th term . Therefore, . Hence or belongs to and thus or .
(2) Suppose that is a prime ideal of . Then is a prime ideal of . Let and be the 0th term of . Then and hence . Since is a prime ideal, . Now, let . Then where is a prime ideal of .
Conversely, note that is an epimorphism. So for any prime ideal of , is a prime ideal of .
(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. Let . (1) is a prime ideal of if and only if is a quasiprime ideal of .(2) is a maximal ideal of if and only if is a quasimaximal ideal of .

Proof. (1)   Obvious.
Suppose that is a prime ideal of . Then for some prime ideal of . But and is an epimorphism. So, .

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

Theorem 4.4. Let be a ring with . Then is a regular ring if and only if is a quasiregular ring.

Proof. By using Corollary 2.10 and Theorem 4.1, it is enough to prove that every prime ideal of is a maximal ideal if and only if every prime ideal of is maximal.
Let be a prime ideal of . Then for some prime ideal of . Hence is a maximal ideal of . Thus is a maximal ideal of .
Conversely, let be a prime ideal of . Then is a quasiprime ideal of . Therefore, is a quasimaximal ideal of and thus is a maximal ideal of .

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

Theorem 4.5. Let be a ring with . Then is a PF-ring if and only if is a quasi-PF ring.

Proof. Note that is prime ideal of if and only if , is prime ideal of . Moreover, is a unique minimal prime ideal of if and only if , is a unique minimal prime ideal contains in .

From Theorems 4.4 and 4.5 we can obtain the following.

Theorem 4.6. Let be a ring with . (1)If is a regular ring and is an ideal of , then is a quasiregular ring.(2)If is a PF-ring and is a pure ideal of , then is a a quasi-PF ring.

Proof. Note that, .

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, is quasi-PF ring but not quasiregular ring since is PF-ring but not regular ring. is a quasi-PF ring but not a quasidomain because 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 that is a quasiregular rings if and only if every differential ideal is pure?

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