International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 376985, 12 pages
http://dx.doi.org/10.1155/2010/376985
Research Article

Note on Isomorphism Theorems of Hyperrings

Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur 628216, Tamilnadu, India

Received 11 May 2010; Revised 19 October 2010; Accepted 29 December 2010

Academic Editor: Heinz Gumm

Copyright © 2010 Muthusamy Velrajan and Arjunan Asokkumar. 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

There are different notions of hyperrings . In this paper, we extend the isomorphism theorems to hyperrings, where the additions and the multiplications are hyperoperations.

1. Introduction

The theory of hyperstructures was introduced in 1934 by Marty  at the Congress of Scandinavian Mathematicians. This theory has been subsequently developed by Corsini , Mittas [5, 6], Stratigopoulos , and by various authors. Basic definitions and propositions about the hyperstructures are found in [3, 4, 8]. Krasner  has studied the notion of hyperfields, hyperrings, and then some researchers, namely, Ameri , Dašić , Davvaz , Gontineac , Massouros , Pianskool et al. , Sen and Dasgupta , Vougiouklis [8, 17], and others followed him.

Hyperrings are essentially rings with approximately modified axioms. There are different notions of hyperrings . If the addition + is a hyperoperation and the multiplication is a binary operation, then the hyperring is called Krasner (additive) hyperring . Rota  introduced a multiplicative hyperring, where + is a binary operation and the multiplication is a hyperoperation. De Salvo  studied hyperrings in which the additions and the multiplications were hyperoperations. These hyperrings were also studied by Barghi  and by Asokkumar and Velrajan . In 2007, Davvaz and Leoreanu-Fotea  published a book titled Hyperring Theory and Applications. Davvaz  extended that the isomorphism theorems to Krasner hyperrings, provided the hyperideals considered in the isomorphism theorems are normal.

In this paper, we extend the isomorphism theorems to hyperrings, in which both the additions and the multiplications are hyperoperations. Also, it is observed that if is an hyperideal of a hyperring and is a normal subcanonical hypergroup of , then is a ring, and hence the quotient hyperrings considered in the isomorphism theorems by Davvaz in  are rings.

2. Basic Definitions and Notations

This section provides some basic definitions that have been used in the sequel. A hyperoperation on a nonempty set is a mapping of into the family of nonempty subsets of (i.e., , for every . The definitions are found in references [3, 4, 8, 24]. A hypergroupoid is a nonempty set equipped with a hyperoperation . For any two nonempty subsets and of a hypergroupoid and for , , we mean the set , , and .

A hypergroupoid is called a semihypergroup if for every (the associative axiom). A hypergroupoid is called a quasihypergroup if for every (the reproductive axiom). A reproductive semihypergroup is called a hypergroup (in the sense of Marty). A comprehensive review of the theory of hypergroups appears in .

Definition 2.1. A nonempty set with a hyperoperation + is said to be a canonical hypergroup if the following conditions hold: (i)for every , ,(ii)for every , ,(iii)there exists (called neutral element of ) such that for all ,(iv)for every , there exists a unique element denoted by such that ,(v)for every , implies and .

Example 2.2. Consider the set . Define a hyperaddition + on as in the following table. Then, is a canonical hypergroup.

The following elementary facts in a canonical hypergroup easily follow from the axioms.(i) for every ,(ii)0 is the unique element such that for every , there is an element with the property ,(iii), (iv) for all .

For any subset of a canonical hypergroup , denotes the set . A nonempty subset of a canonical hypergroup of is called a subcanonical hypergroup of if is a canonical hypergroup under the same hyperoperation as that of . Equivalently, for every , . In particular, for any , . Since , it follows that .

Definition 2.3. An equivalence relation defined on a canonical hypergroup is called strongly regular if for all and implies that for every , for every and for every one has .

Definition 2.4. A subcanonical hypergroup of a canonical hypergroup is said to be normal if for all .

Definition 2.5. The heart of a canonical hypergroup is the union of the sums , where and is a natural number and it is denoted by .

Definition 2.6. Let and be two canonical hypergroups. A mapping from into is called a homomorphism from into if (i) for all and (ii) hold. The mapping is called a good or strong homomorphism if (i) for all and (ii) hold.

A homomorphism (resp., strong homomorphism) from a canonical hypergroup to a canonical hypergroup is called an isomorphism (resp., strong isomorphism) if is one to one and onto. If is strongly isomorphic to , then we denote it by .

Definition 2.7. Let be a homomorphism from canonical hypergroup into a canonical hypergroup . Then, the set is called kernel of and is denoted by , and the set is called Image of and is denoted by .

It is clear that is a subcanonical hypergroup of and is a subcanonical hypergroup of . The definition of a hyperring given below is equivalent to one formulated by De Salvo  (see Corsini ) and studied by Barghi .

Definition 2.8. A hyperring is a triple , where is a nonempty set with a hyperaddition + and a hypermultiplication satisfying the following axioms: (1) is a canonical hypergroup,(2) is a semihypergroup such that for all , (i.e, 0 is a bilaterally absorbing element), (3)The hypermultiplication is distributive with respect to the hyperoperation +. That is, for every , and .
In a hyperring if the hypermultiplication is a binary operation, then it is called as Krasner or additive hyperring. Also, in the Definition 2.8, if the hyperaddition is a binary operation, then it is called as multiplicative hyperring.

Example 2.9. Let be a set with two hyperoperations defined as follows:

Then, is a hyperring.

Definition 2.10. Let be a hyperring, and let be a nonempty subset of . is called a left (resp., right) hyperideal of if is a canonical subhypergroup of and for every and , (resp., ). A hyperideal of is one which is a left as well as a right hyperideal of .

If are left (resp., right) hyperideals of a hyperring , then , are left (resp., right) hyperideal of . If are hyperideals of a hyperring , then , are hyperideals of .

Definition 2.11. Let and be two hyperrings. A mapping from into is called a homomorphism if (i) ; (ii) and (iii) hold for all . The mapping is called a good homomorphism or a strong homomorphism if (i) ; (ii) and (iii) hold for all .

Definition 2.12. A homomorphism (resp., strong homomorphism) from hyperring into a hyperring is said to be an isomorphism (resp., strong isomorphism) if is one to one and onto. If is strongly isomorphic to , then it is denoted by .

Remark 2.13. Let be a homomorphism from a hyperring into a hyperring . Then is a hyperideal of and is a hyperideal of .

3. Canonical Hypergroups

Let be a subcanonical hypergroup of a canonical hypergroup . In this section, we construct quotient canonical hypergroup and prove that when is normal, is an abelian group.

Proposition 3.1. Let be a canonical hypergroup, and let be a subcanonical hypergroup of . For any two elements , if we define a relation if , then ~ is an equivalence relation on .

Proof. Let . Since , the relation ~ is reflexive. Let . If , then for some . That is, . So, ~ is a symmetric relation. Suppose that such that and , then and . Therefore, , and , for some . So, . Hence . Therefore, the relation ~ is transitive.

Remark 3.2. Let be a subcanonical hypergroup of a canonical hypergroup . We denote the equivalence class determined by the element by the equivalence relation ~ by . It is clear that .

Proposition 3.3. Let be a canonical hypergroup, and let be a normal subcanonical hypergroup of . Then, for , the following are equivalent: (1), (2), (3).

Proof. (1) implies (2).
Since , we have . Since is normal subcanonical hypergroup of , we get . Thus, . That is, , and hence .
(2) implies (3) is obvious.
(3) implies  (1). Since , there exists and . Therefore, . If , then . Therefore, . That is, .

Remark 3.4. Let be a canonical hypergroup, and let be a subcanonical hypergroup of . When is normal, the equivalence relation defined in the Proposition 3.1 coincides with the the equivalence relation defined by Davvaz . Further, the Propositions 3.1 and 3.3 are true when the hyperaddition on the canonical hypergroup is not commutative. Also, for any , we have .

Theorem 3.5. Let be a canonical hypergroup, be a subcanonical hypergroup of . Then for , the sets , and are equal.

Proof. Let . Then . Since and we have . Thus . Suppose , then . That is, for some and . Therefore , where . Since , we get . Thus . Hence .
If , then . Therefore, . Hence . On the other hand if , then . Since , we get for some and . Thus . Since , we get . Hence .

Remark 3.6. Let be a canonical hypergroup, and let be a subcanonical hypergroup of . Then, we denote the collection of all equivalence classes induced by the equivalence relation ~ by .

Theorem 3.7. Let be a canonical hypergroup, and let be a subcanonical hypergroup of . If we define for all , then is a canonical hypergroup.

Proof. If such that and , then and . Let . Since is commutative, for some and for some . That is, . Hence, . Also, since and , by a similar argument, we get, . Hence, . Thus, hyperaddition is well defined.
Let . If , then for some . That is, for some . Also, for some . Now, . That is, for some . So, for some . This means that and . Since , we have . This means that . Hence . Similarly, we get . Hence, . Thus, the hyperaddition is associative.
Consider the element . Now, for any , we have . Similarly, . Thus, is the zero element of .
Let , then . Since , we get . Similarly, we can show that . Let , and suppose that is such that , then , where . That is, , and hence . Thus, the element has a unique inverse .
Suppose that , then , where . This implies . That is, , where . Thus, . Similarly, we can show . Since is commutative, it is obvious that is also commutative. Thus, is a canonical hypergroup.

Corollary 3.8. Let be a strong homomorphism from canonical hypergroup into a canonical hypergroup , then is a canonical hypergroup.

Remark 3.9. Let be a canonical hypergroup, and let be a subcanonical hypergroup of . We denote the subset of by .

Proposition 3.10. Let be a canonical hypergroup, and let be a subcanonical hypergroup of . Then, is a subcanonical hypergroup of containing .

Proof. Let . Since is a subcanonical hypergroup of , . This implies . Therefore, . Since , the set is nonempty.
Let . For , we get . Hence, . That is, . Therefore, is a subcanonical hypergroup of containing .

Definition 3.11. Let be a canonical hypergroup, and let be a subcanonical hypergroup of .   is called a subgroup of if is a group. That is, is a singleton set for all .

Example 3.12. The set with the following hyperoperation + is a canonical hypergroup

In this example are subgroups of and whereas in the Example 2.2, is the subgroup of and .

Proposition 3.13. Let be a canonical hypergroup. Then, is the subgroup of containing all subgroups of .

Proof. By the Proposition 3.10, is the subcanonical hypergroup of . Let . Consider the set . If , then . Hence, . This means that the set has only one element. Thus, is a subgroup of . Suppose, is any subgroup of , then for any that we have . That is, . Hence, . Thus, contains all subgroups of .

Corollary 3.14. Let be a canonical hypergroup. Then, is an abelian group if and only if .

Proposition 3.15. Let be a canonical hypergroup, and let be a subcanonical hypergroup of . Then, is normal if and only if .

Proof. Let be normal. Then, for , . That is, . Hence, . Conversely, if , then for , we get . Thus, is normal.

Proposition 3.16. The heart of a canonical hypergroup is a normal subcanonical hypergroup of .

Proof. If , then and , where and are natural numbers. Thus . Now, for any element , there exists natural number and elements such that . Then, for any , . Hence, heart is a normal subcanonical hypergroup of .

Proposition 3.17. A subcanonical hypergroup of a canonical hypergroup is normal if and only if contains the heart of the canonical hypergroup .

Proof. Let be a normal subcanonical hypergroup of the canonical hypergroup . Then for every , and . In particular, when , we get for every . Since is a subcanonical hypergroup of , the union of the sums for and is a natural number. That is, . Conversely, assume that subcanonical hypergroup contains the heart of the canonical hypergroup . For and , . Hence, is a normal subcanonical hypergroup.

From Propositions 3.16 and 3.17, we have the following proposition.

Proposition 3.18. In a canonical hypergroup , is the smallest normal subcanonical hypergroup.

Proposition 3.19. Let be subcanonical hypergroups of a canonical hypergroup such that , then .

Proof. Let . Then, . That is, . Hence, .

Proposition 3.20. Let be subcanonical hypergroups of a canonical hypergroup such that . If is normal, then is also normal.

Proof. If is normal, then by Proposition 3.15, . Since , by Proposition 3.19, . Hence, . By Proposition 3.15, is normal.

Corollary 3.21. Let be subcanonical hypergroups of a canonical hypergroup such that is normal, then the subcanonical hypergroup is also normal.

Corollary 3.22. Let be a canonical hypergroup such that is normal, then all the subcanonical hypergroups are normal.

Theorem 3.23. Let be a canonical hypergroup. Then, the following are equivalent: (i) is an abelian group,(ii) is a normal subcanonical hypergroup of ,(iii).

Proof. By Corollary 3.14, a canonical hypergroup is an abelian group if and only if . By Proposition 3.15, if and only if is a normal subcanonical hypergroup of . Hence, a canonical hypergroup is an abelian group if and only if is a normal subcanonical hypergroup of .
By Proposition 3.18, is the smallest normal subcanonical hypergroup of . Therefore, is normal if and only if .

Corollary 3.24. is an abelian group if and only if all subcanonical hypergroups of are normal.

Theorem 3.25. Let be a canonical hypergroup, and let be a normal subcanonical hypergroup of . Then, is an abelian group.

Proof. For the quotient canonical hypergroup , the zero element is . Since for all , we have is a normal subcanonical hypergroup in . By Theorem 3.23, is an abelian group.

Remark 3.26. If is a normal subcanonical hypergroup of a canonical hypergroup , then the relation ~ defined in Proposition 3.1, is a strongly regular equivalence relation. Hence, by Theorem  31 in , is an abelian group. However, we have proved Theorem 3.25 in a different way.

4. Isomorphism Theorems of Canonical Hypergroups

In this section, we prove the isomorphism theorems of canonical hypergroups.

Theorem 4.1 (First Isomorphism Theorem). Let be a strong homomorphism from a canonical hypergroup into a canonical hypergroup with kernel . Then, is strongly isomorphic to .

Proof. Define a map by for all . Suppose that , where , then . That is, for some . Hence, . So . Hence, . Thus, the map is well defined.
If , then Also, Thus, . Moreover, . Hence, is a strong homomorphism.
Suppose that such that , then . This means that . That is, for some . Since , we get . Now, . Then, by Proposition 3.3   and hence is one to one. Clearly, is onto. Thus, is a strong isomorphism. That is, is strongly isomorphic to .

Corollary 4.2. Let be a strong homomorphism from a canonical hypergroup onto a canonical hypergroup with kernel . Then, is isomorphic to .

Theorem 4.3 (Second Isomorphism Theorem). If and are subcanonical hypergroups of a canonical hypergroup , then .

Proof. It is clear that we can consider the subcanonical hypergroup of the canonical hypergroup as a canonical hypergroup for which is a subcanonical hypergroup. Similarly, the subcanonical hypergroup of the canonical hypergroup as a canonical hypergroup for which is a subcanonical hypergroup.
Define by for every . For all , . Moreover, . Thus, is a strong homomorphism.
Now, implies that for some . That is, for some , . Since , we get . Thus, . Thus, is onto. Let . Then, . Thus, if and only if . Hence, by the First Isomorphism Theorem, .

Theorem 4.4 (Third Isomorphism Theorem). If and are subcanonical hypergroup of a canonical hypergroup such that , then .

Proof. Define a map by . Then, is a strong onto homomorphism of canonical hypergroup with kernel . Therefore, by the First Isomorphism Theorem of canonical hypergroups, .

5. Isomorphism Theorems of Hyperrings

Let be a hyperring, and let be a hyperideal of . Since is a subcanonical hypergroup of is a canonical hypergroup under the hyperaddition defined in the Theorem 3.7. In this section, we define a hypermultiplication on and prove that is a hyperring.

Theorem 5.1. If we define for all , then is a hyperring.

Proof. If such that and , then and . Let . Then, for some and for some . That is, and so . Similarly, we get, . Hence, . Thus, hypermultiplication is well defined.
Suppose, . Then, Thus, we get . Hence, hypermultiplication is associative. Further, Also, Hence, . Similarly, we can show that . Therefore, hypermultiplication is distributive with respect to the hyperaddition. Thus, is a hyperring.

Corollary 5.2. Let be a strong homomorphism from hyperring into a hyperring , then is a hyperring.

Remark 5.3. If is a Krasner hyperring and is a hyperideal of , then is also a Krasner hyperring. Further if is a normal subcanonical hypergroup of , then by the Theorems 3.23 and 5.1, is a ring. Hence, the quotient hyperrings considered in  are just rings. So, in the isomorphism theorems proved in , all the quotient hyperrings considered are rings. However, we prove the isomorphism theorems of hyperrings in which the additions and the multiplications are hyperoperations.
If is a hyperring, and is a hyperideal of , and is a normal subcanonical hypergroup of , then is a multiplicative hyperring.

Theorem 5.4 (First Isomorphism Theorem). Let be a strong homomorphism from a hyperring into a hyperring with kernel . Then, is strongly isomorphic to .

Proof. Define a map by for all .
By Theorem 4.1, this map is a strong isomorphism from canonical hypergroup onto . Now,
Thus, . Hence, is a strong hyperring isomorphism.

Corollary 5.5. Let be a strong homomorphism from a hyperring onto a hyperring with kernel . Then, is strongly isomorphic to .

Theorem 5.6 (Second Isomorphism Theorem). If and are hyperideals of a hyperring , then .

Proof. We can consider the hyperideal of the hyperring as a hyperring for which is a hyperideal. Similarly, hyperideal of the hyperring as a hyperring for which is a hyperideal.
Define by for every . By Theorem 4.3, is strong isomorphism from canonical hypergroup onto the canonical hypergroup . Now, . Thus, is strong isomorphism from hyperring onto the hyperring . Also, from Theorem 4.3,. Hence, by First Isomorphism Theorem of hyperrings, .

Theorem 5.7 (Third Isomorphism Theorem). If and are hyperideals of a hyperring such that , then .

Proof. Define a map by . Then, is a strong onto homomorphism of hyperring with kernel . Therefore, by the First Isomorphism Theorem of hyperrings, .

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