-Primary Hyperideals on Commutative Hyperrings

Abstract

The purpose of this paper is to define the hyperideal expansion. Hyperideal expansion is associated with prime hyperideals and primary hyperideals. Then, we define some of their properties. Prime and primary hyperideals’ numerous results can be extended into expansions.

1. Introduction

The hyperstructure theory was introduced by Marty (1934). Hyperstructures have many applications to several sectors of both pure and applied mathematics. A hypergroup in the sense of Marty is a nonempty set endowed by a hyperoperation [1], the set of the entire nonempty set , which satisfies the associative law and reproduction axiom. Canonical hypergroups are a special class of the hypergroup of Marty. The more general structure that satisfies the ring-like axioms is the hyperring in the general sense: is a hyperring if + and are two hyperoperations such that is a hypergroup and is an associative hyperoperation, which is distributive with respect to +. There are different notions of hyperrings [1]. If only the addition + is hyperoperation and the multiplication is usual operation, then we say that is an additive hyperring. A special case of this type is the hyperring introduced by Krasner (1957) [2]. Also, Krasner (1983) introduced a class of hyperring and hyperfields and the quotient hyperrings and hyperfields. If only is a hyperoperation, we shall say that is a multiplicative hyperring [2]. Rota (1982) introduced the multiplicative hyperrings; subsequently, many authors worked on this field (Nakassis, 1988; Olson and Ward, 1997; Procesi and Rota, 1999; Rota, 1996) [2]. Algebraic hyperstructures have been studied in the following decades and nowadays by many mathematicians.

Although the -primary ideals have been investigated by Dongsheng [3], the concept of -primary hyperideals which unify prime hyperideals and primary hyperideals has not been studied yet. So, this work shows some elementary properties of the hyperideal expansion; then we show some new results of -primary hyperideals. After this introductory section, Section 2 is devoted to some definitions and properties related to primary ideals and hyperideals that will be needed later. In Section 3, the definitions of hyperideal expansion and primary hyperideals will be given and some basic properties of these concepts will be studied.

2. Preliminaries

Throughout this paper denotes the Krasner hyperring.

Definition 1 (see [4]). A Krasner hyperring is an algebraic structure which satisfies the following axioms:(1) is a canonical hypergroup; that is,(i)for every , ,(ii)for every , ,(iii)there exists such that for every ,(iv)for every there exists a unique element such that ,(v) implies and ,(2) is a semigroup having zero as a bilaterally absorbing element; that is, .(3)The multiplication is distributive with respect to the hyperoperation +.

Definition 2 (see [2]). Let be a hyperring and be a nonempty subset of . Then is said to be a subhyperring of if is itself a hyperring.

Definition 3 (see [1]). A subhyperring of a hyperring is a left (right) hyperideal of if () for all and . is called a hyperideal if is both a left and a right hyperideal.

Lemma 4 (see [2]). A nonempty subset of a hyperring is a hyperideal if and only if(1) implies ,(2) and imply .

Definition 5 (see [2]). Let and be hyperrings. A mapping from into is said to be a good (strong) homomorphism if, for all ,

Definition 6 (see [1]). Let be a hyperring homomorphism. The kernel of , denoted , is the set of elements of that map to in ; that is, .

Definition 7 (see [2]). A hyperideal of a hyperring is called a prime hyperideal if whenever , either or .

Definition 8 (see [2]). Let be a hyperideal of the hyperring . Then the radical of , denoted by , is defined as for some .

Definition 9 (see [2]). A hyperideal of a hyperring is called a primary hyperideal if whenever , either or .

3. Hyperideal Expansion and Primary Hyperideals

Definition 10. An expansion of hyperideals, or briefly hyperideal expansion, is a function which assigns to each hyperideal of a hyperring another hyperideal of the same ring such that the following conditions are satisfied: (i).(ii) implies for hyperideals of .

Example 11. Let denote the set of all hyperideals of the hyperring . The identity function , where for every , is an expansion of hyperideals.
For each hyperideal define , the radical of . Then is an expansion of hyperideals.

Definition 12. Given an expansion of hyperideals, a hyperideal of is called -primary if and imply for all .
Obviously the definition of -primary hyperideals can be also stated as and implies for all .

Example 13.  
(1) A Hyperideal Is -Primary If and Only If It Is Prime. Let be -primary hyperideal. We show that is prime. Assume that and that is -primary or so is a prime hyperideal.
Conversely, let be a prime hyperideal. Assume that . Since is prime or and is -primary.
(2) A Hyperideal is -Primary If and Only If It Is Primary. Let be -primary. We show that is primary. Assume that . Since is -primary then we can say that or . That is or . So is primary.
Conversely let be a primary hyperideal. Assume that . Since primary hyperideal or , thus or .

Remark 14. (1) If and are two hyperideal expansions and for each hyperideal , then every -primary hyperideal is also -primary. Thus, in particular, a prime hyperideal is -primary for every hyperideal expansion. Let be -primary. Assume that, for all , and . Since is -primary and and , thus .
(2) Given two hyperideal expansions and , define . Then is also a hyperideal expansion. Since and are hyperideal expansions and , then and .
Let and be any hyperideals of and . Thus and . Finally we find .
(3) Let be a hyperideal expansion. Define . Then is still a hyperideal expansion.
For all , we show that for any , if ; then . By the definition of , we conclude that . For any , if , then the -primary hyperideals which contain contain also . In addition, there may be -primary hyperideals which contained but did not contain . Hence, we conclude that .

Lemma 15. A hyperideal is -primary if and only if for any two hyperideals and , if and then .

Proof. Let be -primary. Suppose and , but , and then we can choose and . Then but and . This contradicts the assumption that is -primary.
Conversely, if the condition is satisfied, for any two elements and , suppose and . Then and . So . Hence implies . Thus is -primary.

Recall that if and are ideals of a commutative ring , then their ideal quotient denotes () defined by . We recall also ideal quotient () is itself an ideal in .

Theorem 16. Let be a hyperideal expansion. Then (1)if is a -primary hyperideal and is a hyperideal with , then ,(2)for any -primary hyperideal and any subset of the , is also -primary.

Proof. (1) From the definition of , for all , , and . Since is a hyperideal , then we get . In other words . Since is -primary, if then .
Conversely, since then .
(2) For all , assume that and . Then there exists a such that . But . Thus . Since , . By this way we get that is -primary.

Theorem 17. If is a hyperideal expansion such that for every hyperideal , then, for any -primary hyperideal , .

Proof. For all hyperideals, since , .
Conversely, let . We show that .
Then there exists which is the least positive integer with . If then . If then . But , so . Hence and .

4. Expansions with Extra Properties

In this section we investigate -primary hyperideals where satisfy additional conditions and prove more results with respect to such expansions.

Definition 18. A hyperideal expansion is intersection preserving if it satisfies An expansion is said to be global if for any hyperring homomorphism The expansions and are both intersection preserving and global.

For any , :

And .

. Thus and are both intersection preserving and global.

Theorem 19. Let be an intersection preserving hyperideal expansion. If are -primary hyperideals of and for all , then is -primary.

Proof. Let be an intersection preserving hyperideal expansion. If and , then for some . But and is -primary, so . But . Thus . So is -primary.

Definition 20. Let be a hyperring and be a hyperideal expansion. If for an , then is called nilpotent.

Note that nilpotent element of a ring is the zero element of the ring. Also nilpotent elements are exactly the ordinary nilpotent elements.

Theorem 21. Let be a global expansion. Let a hyperideal of be -primary and then every zero divisor of the quotient hyperring is nilpotent.

Proof. Let be -primary. If is a zero divisor of , then there is a with . This means that and . Since is -primary so ; that is, .
Let be the natural quotient hyperring homomorphism. As is global, we have .
Since is onto, so . Hence we get , so is nilpotent.

Theorem 22. If is global and is a hyperring homomorphism, then, for any -primary hyperideal of , is a -primary hyperideal of .

Proof. Let with . If then but . So, as is -primary, . So . Hence is -primary.

Theorem 23. Let be a surjective hyperring homomorphism. Then a hyperideal of that contains is -primary hyperideal of .

Proof. If is -primary, then by and Theorem 22, is -primary. Now suppose is -primary. If and and , then there are with . Then implies and implies and implies .
So , and hence . Now one only needs to prove . But this follows directly from and that is surjective.

The following theorem does not need a proof because it is a consequence of Theorems 22 and 23.

Correspondence Theorem for  -Primary Hyperideals. Let be a hyperring homomorphism of a hyperring onto a hyperring and let be global hyperring expansion. Then induces a one-one inclusion preserving correspondence between -primary hyperideals of containing and the -primary hyperideals of in such a way that if is a -primary hyperideal of that contains , then is the corresponding -primary hyperideal of , and if is a -primary hyperideal of , then is the corresponding -primary hyperideal of .

Conflicts of Interest

The authors declare that they have no conflicts of interest.