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Advances in Fuzzy Systems
Volume 2010, Article ID 258947, 4 pages
http://dx.doi.org/10.1155/2010/258947
Research Article

Two New Types of Rings Defined by Using a Translational Invariant Fuzzy Subset

Department of Mathematics, Irbid National University, P.O. Box 2600, Irbid 21110, Jordan

Received 4 March 2010; Revised 16 June 2010; Accepted 10 August 2010

Academic Editor: Eyke Huellermeier

Copyright © 2010 Manal Ghanem. 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

We use a translational invariant fuzzy subset of a ring to define two new types of commutative rings namely, -presimplifiable and -associate rings. We present some results of these rings. The interest of these results is that most of them are mirrors of corresponding results of presimplifiable and associate rings in classical ring theory.

1. Introduction

Throughout this paper, all rings are assumed to be commutative with unity. If is a ring, then ,    and denote the Jacobson radical of , the nilradical of , the set of zero divisors of , and the set of units of , respectively.

Recall that a ring is presimplifiable if whenever with property we have or . And, a ring is associate if whenever satisfy and for some there is a unit such that . The class of associate rings contains a large class of rings such as presimplifiable rings, Principal ideal rings, Artinian rings, regular rings, -rings, and stable range one rings. This class of rings was originally studied in Kaplansky [1]. Then Bouvier studied presimplifiable rings in a series of papers [25]. Recently, the class of associate rings was studied more extensively by Anderson and Valdes-Leon [6, 7], Spellman et al. [8], and Anderson et al. [9].

Some well-known properties of presimplifiable and associate rings will be given in the following.

Remark 1. (1) A ring is presimplifiable if and only if ; see Anderson et al. [9].
(2) A homomorphic image of a presimplifiable ring need not be associate; see Spellman et al. [8].
(3) Integral domains, domain-like rings (i.e., ) and local rings are examples of presimplifiable rings; see Anderson et al. [9].
(4) If and are integral domains and is a ring with epimorphisms ,   , which are not isomorphisms, then the pullback P of is presimplifiable if and only if see Anderson et al. [9].
(5) presimplifiable, regular rings are examples of associate rings; see Anderson et al. [9].

Algebraic structures play a prominent role in mathematics with wide ranging applications in many areas such as topological spaces, theoretical physics, coding theory, and computer sciences. This provides sufficient motivation to researchers to extend various concepts and results from the realm of abstract algebra to the broader framework of fuzzy setting, although not all results in algebra can be fuzzified. The concept of a fuzzy subset of a set was introduced by Zadeh [10]. Fuzzy subgroup and their important properties were defined and established by Rosenfeld [11]. Fuzzy ideal of rings was introduced by Liu [12]. The notion of translational invariant fuzzy subset was introduced by Ray in [13]. This notion gave immense scope for extending the classical results of different algebraic structures. Our aim of this paper is to extend the classical results of presimplifiable and associate rings to the fuzzy setting.

Recall that a fuzzy subset of a set is a mapping from into the closed unit interval . A fuzzy subset is called translational invariant with respect to a binary operation “ ” if it satisfies the following condition: implies that for every . An element with is called a -unit of a ring if there exists with such that . If is a translational invariant fuzzy subset of with respect to both “ ” and “·” and , then it is easy to prove that every unit is a -unit. And is -unit if and only if and are -units.

Now, we define two new types of commutative rings by using a translational invariant fuzzy subset of a ring.

Definition 2. Let be a ring and a translational invariant fuzzy subset of with respect to both “ ” and “ ” satisfying and for every .
(1) is said to be a -presimplifiable if whenever with property we have or is a -unit.
(2) is said to be a -associate if whenever satisfy and for some implies that there is a -unit such that .

It is clear that the concepts of -presimplifiable and -associate rings are generalization of the presimplifiable and the associate rings in classical ring theory.

In this paper, in section two, we will study some properties of -presimplifiable and -associate rings and we will show that most of them are very close to that of presimplifiable and associate rings in classical ring theory. In section three, unlike the classical ring case, we will show the structure preserving nature of -presimplifiable and -associate rings under ring epimorphism.

Remark 3. In the next two sections, is assumed to be a translational invariant fuzzy subset of with respect to both “ ” and “ ” satisfying and for every .

2. -Presimplifiable and -Associate Rings

Our aim of this section is to study some properties of -presimplifiable and -associate rings.

Next, we give a characterization of -presimplifiable rings but first we need to state the following.

Definition 4. An element is said to be a -zero if .

Definition 5 (Ray and Ali [14]). An element with is said to be a -divisor of zero if there exists with such that .

The proof of the following lemma is straightforward.

Lemma 6. The set is a -divisor of zero or is a -zero is containing the set .
The set for some is an ideal containing .
The set is a -unit for every is an ideal of containing .

Theorem 7. The following statements are equivalent.(1) is a -presimplifiable ring.(2) .

Proof. (1) (2) Suppose that R is -presimplifiable ring. Let then there exists such that and . But for every because is a translational invariant. So, is a -unit and hence .
(2) (1) Suppose that such that and . Then . Hence or . Therefore, is a -unit of . Thus is -presimplifiable.

Definition 8. (1) A ring is said to be a -integral domain if has no -divisor of zero.
(2) A ring is said to be a -domain-like if .

Clearly, every -integral domain is -domain-like. However, the converse need not be true for example, with a fuzzy subset defined by , , and is -domain-like but not -integral domain.

Theorem 9. Every -domain-like ring is -presimplifiable.

Proof. It is enough to show that . So, let then for some . Hence for every because is translational invariant. Thus is a -unit. So .

Definition 10. A ring is said to be a -local if for every with it follows that or is a -unit.

Example 11. (1) Let and a fuzzy subset of defined by if is even and if is odd; then is -local ring.
(2) Let and a fuzzy subset of defined by , . Then is -local ring.

Theorem 12. Every -local ring is -presimplifiable.

Proof. Note that, is the set of all non- -units of .

Now, we consider the -pullback.

Definition 13. Let , , and be any three rings with homomorphisms ,   , which preserves the unity. Let    be fuzzy subsets of . If for every and satisfying we have that then the set with the fuzzy subset of P defined by for every is a subring of called a -pullback of with the set of -units and are -units .

Theorem 14. Let be a -presimplifiable, a -presimplifiable, and a ring with epimorphisms    . If for every and satisfying we have that then the -pullback of is -presimplifiable.

Proof. Let such that and for some . Then , and . But is -presimplifiable and is -presimplifiable. Thus .

Example 15. (1) Let be a fuzzy subset defined on by and . And be a fuzzy subset defined on by and . Let be the epimorphisms map to . Then the -pullback of is -presimplifiable.
(2) Let for . Let be a fuzzy subset defined on by , and . If and , then the -pullback of is -presimplifiable.

We end this section by studying some properties of -associate rings. Recall that, a ring is -associate if which satisfy and for some implies that there is a -unit such that . It is clear that every -presimplifiable ring is -associate while the converse is not necessarily true as we shall see (Example 18).

Definition 16. A ring is said to be a -Boolean if for all .

Clearly, every Boolean ring is -Boolean but the converse need not be true for example, if is a fuzzy subset of defined by if is even and if is odd, then the ring is -Boolean but it is not Boolean.

Theorem 17. Every -Boolean ring is -associate.

Proof. Let such that and for some . Then and since is a translational invariant fuzzy subset of . So, .

Example 18. Let and a fuzzy subset of defined as follows and . Then is -Boolean and hence it is -associate. However, is not -presimplifiable since and is not a -unit of .

Definition 19. A ring is said to be a -stable range one if for every satisfying implies the existence of an element and a -unit such that .

Theorem 20. Every -stable range one ring is -associate.

Proof. Suppose that is -stable range one and such that and for some . Then . So, there exist an element and a -unit of such that . So, and hence is -associate.

Definition 21. A ring is said to be a -regular if for any there exist a -idempotent (i.e., ) and a -unit such that .

Theorem 22. Every -regular ring is -associate.

Proof. Let be two -idempotents and two -units such that and for some . Then and . Hence and . But and are -units so, and . Therefore, . Then implies that in . Hence in . But is a -unit in so, is a -unit in . Thus for some . So, . Hence, is a -unit of and . So the result holds.

3. Images and Inverse Image under Homomorphism

In this section, we study -presimplifiable and -associate properties under ring homomorphism.

Recall that if is a function from a ring into a ring then a fuzzy subset of is -invariant if implies that for every . Ray and Ali [14] proved that if is an epimorphism then a translational invariant of a fuzzy subset of implies a translational invariant of a fuzzy subset of , where for every . Also, the inverse image of a translational invariant fuzzy subset of is a translational invariant fuzzy subset of , where for every .

It is easy to prove the following lemma.

Lemma 23. Let be a ring epimorphism, an -invariant fuzzy subset of and be a fuzzy subset of .(1) if and only if .(2) is a -unit of if and only if is an -unit of .(3)For every , if and only if .(4) if and only if .(5) is an -unit of if and only if is a -unit of .(6) if and only if .

Theorem 24. Let be an epimorphism and an -invariant fuzzy subset of .(1) is -presimplifiable if and only if is -presimplifiable.(2) is -associate if and only if is -associate.

Proof. (1) Let and be two non- -zero elements of such that . Then and because is an -invariant. But is -presimplifiable, so is a -unit and hence is an -unit of .
Conversely, let and be two non- -zero elements of such that . Then and . But is -unit of because is -presimplifiable. Hence is a -unit of .

Corollary 25. Let be the natural homomorphism and a fuzzy subset of R satisfying for every .(1) is -presimplifiable if and only if is -presimplifiable.(2) is -associate if and only if is -associate.

Proof. Note that, if for every , then is an -invariant fuzzy subset of .

Corollary 26. Let be a -Boolean ring or -integral domain ring and the natural homomorphism.(1) is -presimplifiable if and only if is -presimplifiable.(2) is -associate if and only if is -associate.

Theorem 27. Let be an epimorphism and a translational invariant fuzzy subset of .(1) is -presimplifiable if and only if is -presimplifiable.(2) is -associate if and only if is -associate.

Proof. (2) Suppose that satisfy and for some . Then and . But is -associate, so there exists an -unit satisfying . Hence is a -unit of satisfying . Thus is -associate.

Corollary 28. Let be the natural homomorphism and a translational invariant fuzzy subset of .(1) is -presimplifiable if and only if is -presimplifiable.(2) is -associate if and only if is -associate.

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