Journal of Mathematics

Volume 2014 (2014), Article ID 458603, 7 pages

http://dx.doi.org/10.1155/2014/458603

Research Article

Associate, Hyperdomainlike, and Presimplifiable Hyperrings

Agboola Adesina Abdul Akeem¹ and Davvaz Bijan²

¹Department of Mathematics, Federal University of Agriculture, PMB 2240, Abeokuta, Ogun State, Nigeria

²Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran

Received 12 February 2014; Accepted 17 June 2014; Published 14 August 2014

Academic Editor: Feng Feng

Copyright © 2014 Agboola Adesina Abdul Akeem and Davvaz Bijan. 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.

Based on the works of Axtell et al., Anderson et al., and Ghanem on associate, domainlike, and presimplifiable rings, we introduce new hyperrings called associate, hyperdomainlike, and presimplifiable hyperrings. Some elementary properties of these new hyperrings and their relationships are presented.

1. Introduction

The study of strongly associate rings began with Kaplansky in [1] and was further studied in [2–5]. Domainlike rings and their properties were presented by Axtell et al. in [6]. Presimplifiable rings were introduced by Bouvier in the series of papers [7–11] and were later studied in [2–4]. Further properties of associate and presimplifiable rings were recently presented by Ghanem in [12].

The theory of hyperstructures was introduced in 1934 by Marty [13] at the 8th Congress of Scandinavian Mathematicians. Introduction of the theory has caught the attention and interest of many mathematicians and the theory is now spreading like wild fire. The notion of canonical hypergroups was introduced by Mittas [14]. Some further contributions to the theory can be found in [15–19].

Hyperrings are essentially rings with approximately modified axioms. Hyperrings are of different types introduced by different researchers. Krasner [20] introduced a type of hyperring where + is a hyperoperation and is an ordinary binary operation. Such a hyperring is called a Krasner hyperring. Rota in [21] introduced a type of hyperring where + is an ordinary binary operation and is a hyperoperation. Such a hyperring is called a multiplicative hyperring. de Salvo [22] introduced and studied a type of hyperring where + and are hyperoperations. The most comprehensive reference for hyperrings is Davvaz and Leoreanu-Fotea’s book [18]. Some other references are [23–31].

In this paper, we present and study associate, hyperdomainlike, and presimplifiable hyperrings. The relationships between these new hyperrings are presented.

2. Preliminaries

In this section, we will provide some definitions that will be used in the sequel. For full details about associate, domainlike, and presimplifiable rings, the reader should see [1, 4–6, 12]. Also, for details about hyperstructures and hyperrings, the reader should see [12].

Definition 1. Let be a commutative ring with unity.

(1) is called an associate ring if whenever any two elements generate the same principal ideal of , there is a unit such that .(2) is called a domainlike ring if all zero divisors of are nilpotent.(3) is called a presimplifiable ring if, for any two elements with , we have or .(4) is called a superassociate ring if every subring of is associate.(5) is called a superpresimplifiable ring if every subring of is presimplifiable.

Example 2. Integral domains, domainlike, and local rings are presimplifiable rings and hence are associate rings.

Example 3 (see [6]). It is easy to check that is presimplifiable if and only if , where is some prime. Hence, is presimplifiable if and only if is local. Also, if a ring is quasilocal, then is presimplifiable, since , the unique maximal ideal of , and thus .

Definition 4. Let be a nonempty set and let be a hyperoperation, where is the family of all nonempty subsets of . The couple is called a hypergroupoid. For any two nonempty subsets and of and , one defines

Definition 5. A hypergroupoid is called a semihypergroup if for all of one has , which means that

A hypergroupoid is called a quasihypergroup if for all of one has . This condition is also called the reproduction axiom.

A hypergroupoid which is both a semihypergroup and a quasihypergroup is called a hypergroup.

Definition 6. Let be a nonempty set and let + be a hyperoperation on . The couple is called a canonical hypergroup if the following conditions hold:

(1), for all ;(2), for all ;(3)there exists a neutral element such that , for all ;(4)for every , there exists a unique element such that ;(5) implies and , for all .

Definition 7. Let be a nonempty set and let + and be hyperoperation and usual operation on , respectively. The triple is called a Krasner hyperring if the following axioms hold:

(1) is a canonical hypergroup;(2) is a semigroup having zero as a bilaterally absorbing element; that is, for all ;(3)multiplication is distributive over the hyperoperation +; that is, and for all .

If is a hyperring, and are nonempty subsets of and ; then we define

A hyperring is called a commutative hyperring with unity if is a commutative semigroup with unity.

For any , we use instead of .

3. Associate, Domainlike, and Presimplifiable Hyperrings

Throughout this section, all hyperrings will be assumed to be commutative Krasner hyperrings with unity.

, , , and will denote Jacobson radical, nilradical, the set of units, and the set of zero divisors of , respectively. will denote the annihilator of .

Let be a subset of a hyperring . Let be the family of all hyperideals in which contain . Then, is called the hyperideal generated by . This hyperideal is denoted by . If , then the hyperideal is denoted by .

Lemma 8 (see [18]). Let be a hyperring and . Then,

(1) is a hyperideal of ;(2)if has a unit element, then .

Definition 9. Let be a hyperring and let , , and be relations on defined as follows.

(1) if and only if , for all .(2) if and only if , for all and some .(3) if and only if and when and , then .

Theorem 10. Let be a hyperring. Then,

(1) is an equivalence relation on ,(2) is an equivalence relation on ,(3) is an equivalence relation on if and only if, for all , implies or .

Proof. (1) and (2) are clear. For (3), suppose that is an equivalence on and suppose that with . Then, for some which implies . But then, so that . The converse is straightforward.

Definition 11. If is a hyperring, then in is said to be a zero divisor if there exists in such that .

Definition 12. A hyperring is a hyperdomain if it has no zero devisor.

The following example is presented in [18, 29, 30].

Example 13. Let be a finite group with elements, , and define a hyperaddition and a multiplication on , by

Then, is a hyperring.

Example 14. It is easy to see that Example 13 is a hyperdomain.

Theorem 15. Let be a hyperring.

(1) implies , for all .(2) implies , for all .(3)If is hyperdomain, then implies , for all .

Proof. (1) and (2) are obvious. For (3), suppose that is hyperdomain and suppose that . Then, which implies that and for some . Thus, which implies for some so that . Since is a hyperdomain, if , then , and, therefore, which implies . Hence, .

Definition 16. Let be a hyperring.

(1) is said to be an associate hyperring if implies for all .(2) is said to be a superassociate hyperring if every subhyperring of is associate.

Example 17. The hyperring in Example 13 is associate hyperring and superassociate hyperring too.

Definition 18. Let be a hyperring.

(1) is said to be a presimplifiable hyperring if, for all , implies or .(2) is said to be a superpresimplifiable hyperring if every subhyperring of is presimplifiable.(3) is said to be a hyperdomainlike hyperring if all zero divisors of are nilpotent.

Example 19. Let , , and be three rings, where and is a prime number. Suppose that , , are maximal ideals of , . Let , , be subsets of and define a hyperaddition and multiplication on by

Then, are hyperrings. Since , , are domainlike (see Example 3), consequently each is hyperdomainlike.

Theorem 20. Let be a hyperring. The following conditions are equivalent:

(1) implies , for all ;(2) implies , for all ;(3) for all ;(4) is presimplifiable;(5);(6);(7)for , implies .

Proof. (): it follows from the definitions of , and .

(): suppose that which implies that for all . Then, implies and, therefore, for all .

(): suppose that for all and suppose that . Then, for some which implies . If , we must have . Hence, is presimplifiable.

(): suppose that is presimplifiable and suppose that . Then, there exists such that . But then, which implies that for some so that .

(): suppose that . Then, for all . Thus, for some . Now, implies and, therefore, .

(): suppose that and . Then, for some so that for some . Since , we must have so that and, therefore, .

(): suppose that and . Then, and for some . Thus, and we have which implies . Hence, .

Theorem 21. Let be a hyperring. If is quasilocal, then it is presimplifiable.

Proof. Suppose that is quasilocal. Then, has a unique maximal hyperideal and . Hence, and therefore is presimplifiable.

Theorem 22. Let be a hyperring. If is presimplifiable, then is associate.

Proof. It follows from Theorem 20.

Corollary 23. Any hyperdomain or any quasilocal hyperring is an associate hyperring.

Proof. This is immediate from Theorems 20, 21, and 22.

Theorem 24. Let be a hyperring. Then, the following statements are equivalent:

(1) is hyperdomainlike;(2);(3) is a primary hyperideal of .

Proof. (): it is obvious.

(): let . Then, . If , then so that for some positive integer and, therefore, . Hence, is primary.

(3) implies (1): suppose that is a primary hyperideal of . Let be arbitrary. Then, there exists such that which implies that . Since , we must have and therefore . Hence, is nilpotent. Since is arbitrary, it follows that is hyperdomainlike.

Let and be any two hyperrings and let . For any , define

Then, is a hyperring called the direct product of and .

Theorem 25. Let be a hyperring.

(1)If is hyperdomainlike, then it is superpresimplifiable.(2)If is presimplifiable (associate), then a subhyperring of need not be presimplifiable (associate).(3)If is superassociate, then it need not be presimplifiable.(4)A direct product of superassociate hyperrings need not be superassociate.

Proof. See Anderson et al. [4].

Theorem 26. Let and be any two presimplifiable hyperrings with the set of units . Then, is a superassociate hyperring.

Proof. The proof is similar to the proof of Theorem  2.1 in [12].

Theorem 27. Let be a hyperideal of a hyperring . is primary if and only if is hyperdomainlike.

Proof. Suppose that is primary. Let . Then, there exists such that which implies that which implies . Since is primary and , it follows that from which we have and therefore . Hence, and, thus, is hyperdomainlike.

Conversely, suppose that is hyperdomainlike. Since is a zero of , it follows that is primary.

Theorem 28. Let be a hyperring. If is hyperdomainlike, then is presimplifiable.

Proof. The proof follows from Theorem 21, since .

Theorem 29. Let be a hyperring. If is hyperdomainlike, then is a hyperdomain.

Proof. Suppose that is hyperdomainlike. Let such that with . Then, implies but . For some positive integer , we have which implies . Since , we must have for some positive integer . Hence, and, therefore, . Hence, is a hyperdomain.

Theorem 30. Let be a hyperring. If is hyperdomainlike, then is the unique minimal prime hyperideal of .

Proof. Suppose that is hyperdomainlike. Then, . Since is the intersection of all prime hyperideals of , the required result follows.

Theorem 31. Let be a hyperring and let be a hyperideal of . Then, is hyperdomainlike if and only if is a hyperdomain.

Proof. Suppose that is a hyperdomain. Then, is a prime hyperideal of which is clearly a primary hyperideal of . Hence, is hyperdomainlike.

Conversely, suppose that is hyperdomainlike. Let such that with . Then, implies but . Since is primary in , we must have for some positive integer . Thus, for some positive integer . Thus, and, therefore, . Hence, is a hyperdomain.

Theorem 32. Let be a hyperring. is a hyperdomainlike ring if and only is a hyperdomain.

Proof. The proof is similar to the proof of Theorem 29.

Theorem 33. Let be a hyperring and let be the total quotient hyperring of . If is hyperdomainlike so also is .

Proof. The proof is similar to the classical ring and therefore omitted.

Theorem 34. Let be a hyperring. is presimplifiable if and only if whenever and , then .

Proof. Suppose that is presimplifiable and suppose that with . Then, and so that . Now, which implies that for some and therefore and , where and so that . Since , we must have and hence .

Conversely, suppose that with implies that . Then, and . It can be shown that . Hence, is presimplifiable.

Corollary 35. Let be a hyperring. If is presimplifiable, then is presimplifiable and hence strongly associate.

Definition 36. Let be a hyperideal of a hyperring and let . is called a presimplifiable hyperideal if, for every and , implies that .

Lemma 37. Every hyperideal of a presimplifiable hyperring is presimplifiable.

Proof. It is obvious.

Theorem 38. Let be a hyperideal of a hyperring . Then, is presimplifiable if and only if for all .

Proof. Suppose that is presimplifiable. Let , where . Then, and we can write for some and . Since is presimplifiable, we have so that . Hence, .

Conversely, suppose that for all . Let . Then, for some . Thus, and, therefore, so that . Hence, is presimplifiable.

Theorem 39. Let be a hyperring. Then, is presimplifiable if and only if and are presimplifiable.

Proof. Suppose that is presimplifiable. Let . Then, there exits such that so that which implies that and, for a positive integer , we have . Since , we must have and so . Hence, and, therefore, is presimplifiable.

Conversely, suppose that and are presimplifiable. Let such that . Then, . Obviously, and . Consequently, and, therefore, is presimplifiable.

Definition 40. Let and be any two hyperrings and let be a mapping from into .

(1) is called a homomorphism if

(i), for all ,(ii), for all ,(iii).

(2) is called a good or strong homomorphism if

(i), for all ,(ii), for all ,(iii).

(3)A strong homomorphism from a hyperring into a hyperring is called an isomorphism if is bijective and we write .(4)If is a homomorphism from a hyperring into a hyperring , then the kernel of denoted by is the set and the image of denoted by is the set . It is known that is a hyperideal of and is a hyperideal of .

Definition 41. Let , , and be any three hyperrings with strong homomorphisms , , which preserve the unity. The set is called the pullback of .

Lemma 42. Let , , and be any three hyperrings and let be the pullback of . Then, is a subhyperring of and , the set of units of , is .

Theorem 43. Let , , and be any three hyperrings with strong homomorphisms , . If and there exists with , then the pullback of is not an associate hyperring.

Proof. The proof is similar to the proof of Proposition  6 in [4].

Theorem 44. Let , , and be any three hyperrings with strong epimorphisms , , which are not strong isomorphisms. If and are hyperdomains, then the pullback of is associate (presimplifiable) if and only if , respectively, , , or, equivalently, , .

Proof. The proof is similar to the proof of Theorem  7 in [4].

Theorem 45. Let , , and be any three hyperrings with strong epimorphisms , , which are not strong isomorphisms. If and are presimplifiable, then the pullback of is presimplifiable if and only if , .

Proof. The proof is similar to the proof of Theorem  2.3 in [12].

Theorem 46. Let , , and be any three hyperrings with strong epimorphisms , , which are not strong isomorphisms. If and are hyperdomainlikes, then the pullback of is hyperdomainlike if and only if , .

Proof. The proof is similar to the proof of Theorem  2.5 in [12].

Corollary 47. Let , , and be any three hyperrings with strong epimorphisms , , which are not strong isomorphisms. If and are hyperdomainlikes, then the pullback of is associate but not hyperdomainlike if and only if , , and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for their critical reading of the paper and their suggestions.

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