#### Abstract

In this paper, we introduce the concept of 1-absorbing prime hyperideals which is an expansion of the prime hyperideals. Several properties of the hyperideals are provided. For example, it is proved that if a strong C-hyperideal of is 1-absorbing prime that is not prime, then is a local multiplicative hyperring. Moreover, we introduce and study the notions of 1-absorbing primary hyperideals, strongly 1-absorbing primary hyperideals and weakly 1-absorbing primary hyperideals which are generalizations of the 1-absorbing prime hyperideals. We also examine the relations between these new concepts and other hyperideals.

#### 1. Introduction

In a classical algebraic structure, the composition of two elements is an element, but in an algebraic hyperstructure, the composition of two elements is a set. A well-known type of a hyperring is called the Krasner hyperring [1]. The hyperring is a hypercompositional structure where is a canonical hypergroup and is a semigroup in which the zero element is absorbing and the operation is a two-sided distributive one over the hypercomposition . In 1982, Rota initiated the study of multiplicative hyperring [2] which was subsequently investigated by many authors [37]. In the hyperring, the multiplication is a hyperoperation, while the addition is an operation. The hyperring in which the additions and the multiplications are hyperoperations was introduced by Salvo in [8].

In the theory of rings, the key role of the notion of prime ideal as an expansion of the notion of prime number in the ring is undeniable. The notion of primeness of hyperideal in a multiplicative hyperring was conceptualized by Procesi and Rota in [6]. The notions of prime and primary hyperideals in multiplicative hyperrings were fully studied by Dasgupta in [9]. Badawi [10] introduced and studied 2-absorbing ideals and this notion is further generalized by Anderson and Badawi [11, 12]. In [13], Ghiasvand introduced the concept of 2-absorbing hyperideals in a multiplicative hyperring. Several authors have extended this concept in several ways [1418]. The concept of 1-absorbing prime ideals which is another extension of prime ideals was introduced in [19]. The notions of 1-absorbing primary and weakly 1-absorbing primary ideals were investigated in [20].

Motivated from the concept of the prime hyperideals, in this paper, we introduce and study the concept of 1-absorbing prime hyperideals and some of its generalizations in a multiplicative hyperring. Several properties of them are provided.

The paper is organized as follows. In Section 2, we have given some basic definitions and results of multiplicative hyperrings which we need to develop our paper. In Section 3, we introduce the concept of 1-absorbing prime hyperideals and give some basic properties of them. For example we show (Theorem 1) that if is a 1-absorbing prime hyperideal of , then is a prime hyperideal of . Furthermore, is a prime hyperideal of for every nonunit element . In Section 4, we study an expansion of the 1-absorbing prime hyperideals which is called 1-absorbing primary hyperideals. In particular, we discuss the relations between 1-absorbing primay hyperideals and primary hyperideals. In Section 5, the notion of strongly 1-absorbing primay hyperideals is studied. For example, it is shown (Theorem 23) that there exists a strongly 1-absorbing primary hyperideal of if and only if is a prime hyperideal or is a local multiplicative hyperring. In the final section, we investigate the concept of weakly 1-absorbing primary hyperideals of .

#### 2. Preliminaries

Recall first the basic terms and definitions from the hyperring theory. A commutative multiplicative hyperring is an abelian group in which a hyperoperation is defined satisfying the following:(i)for all , we have ;(ii)for all , we have and ;(iii)for all , we have .(iv)for all , we have .

If in (ii) the equality holds then we say that the multiplicative hyperring is strongly distributive. Throughout this paper all hyperrings are commutative multiplicative hyperrings.

Let and be two nonempty subsets of and . Then we define

A non empty subset of is a hyperideal if(i)If , then ;(ii)If and , then .

An element is said to be scalar identity if for all .

Definition 1. (see [21]). A proper hyperideal of is maximal in R if for any hyperideal of with then . Also, we say that is local, if it has just one maximal hyperideal.

Definition 2. (see [9]). A proper hyperideal of is called a prime hyperideal if for implies that or . The intersection of all prime hyperideals of containing is called the prime radical of , being denoted by . If does not have any prime hyperideal containing , we define .
Let . A hyperideal of is said to be a C-hyperideal of if for any implies . Let be a hyperideal of . Then, where . The equality holds when is a C-hyperideal of ([9], Proposition 3.2). In this paper, we assume that all hyperideals are C-hyperideal.
Recall that a hyperideal of is called a strong C-hyperideal if for any , , then , where and (for more details see [22]).

Definition 3. (see [9]). A non-zero proper hyperideal of is called a primary hyperideal if for implies that or . Since is a prime hyperideal of by Proposition 3.6 in [9], is referred to as a P-primary hyperideal of .

Definition 4. (see [23]). Let and be two multiplicative hyperrings. A mapping from into is said to be a good homomorphism if for all , and .

Definition 5. (see [21]). We call as of . Also, for all if and only if .

Definition 6. (see [21]). An element is called unit, if there exists , such that . Denote the set of all unit elements in by .

Definition 7. (see [24]). A hyperring is called an integral hyperdomain, if for all , implies that or .

Definition 8. A hyperring is said to be a reduced hyperring if it has no nilpotent elements. That is, if for and a natural number , then .

Example 1. Consider the ring that for all , and are the remainder of and , respectively, which “” and “” are ordinary addition and multiplication, and . The set with the operation and the hyperoperation for all is a reduced multiplicative hyperring, since for all and all .

Definition 9. A multiplicative hyperring is called a hyperfield if every non-zero element of is unit.

Example 2. Consider the ring for some prime integer . Let and . We define hyperoperation on by for all . Since every non-zero element of is unit, then is a hyperfield.

Definition 10. (see [25]). An element is called regular if there exists such that . So, is called regular, if all of elements in are regular elements. The set of all regular elements in is denoted by .

Definition 11. Let be a hyperideal of and . Then define:

#### 3. 1-Absorbing Prime Hyperideals

Definition 12. Let be a proper hyperideal of . is called a 1-absorbing prime hyperideal if for nonunit elements , , then either or .

Example 3. In the multiplicative hyperring of integers with such that for any , every principal hyperideal generated by a prime integer is a 1-absorbing prime hyperideal.
It is clear that every prime hyperideal of is a 1-absorbing prime hyperideal of . Recall that a hyperideal of is said to be a 2-absorbing primary hyperideal if for , , then or or . Every 1-absorbing prime hyperideal of is a 2-absorbing primary hyperideal of . The following example shows that the converse may not be true, in general.

Example 4. Consider the multiplicative hyperring of integers with such that for any . It is easy to see is a 2-absorbing primary hyperideal of . But it is not a 1-absorbing prime hyperideal of as the fact that but and .

Theorem 1. If is a 1-absorbing prime hyperideal of , then is a prime hyperideal of . Furthermore, is a prime hyperideal of for every nonunit element .

Proof. Suppose that is a 1-absorbing prime hyperideal of and for some nonunit . Therefore for a positive integer . Clearly, we have for some positive integer . Since the hyperideal of is 1-absorbing prime, then we get either or . This means either or . Consequently, the hyperideal of is prime. Now, let for some nonunit elements . Assume that . This means . Since and the hyperideal is 1-absorbing prime, then we get . Consequently, the hyperideal is prime.
The following lemma is needed in the proof of our next result.

Lemma 1. Let be a unit element of for every nonunit element of and for every unit of . Then is local.

Proof. Let and be two different maximal hyperideals of . Hence for some and we have . By Theorem 1 in [21], is a unit element of which means contains a unit element. This is a contradiction. Thus is local.

Theorem 2. If a strong C-hyperideal of is 1-absorbing prime that is not prime, then is local.

Proof. Let be a 1-absorbing prime hyperideal of . Assume that is not prime hyperideal. Suppose that we have for some but . Assume that is a nonunit element and is a unit element of . Let be a nonunit element of . Since and the hyperideal of is 1-absorbing prime, then we get . Since and , then we have . Since and is a strong C-hyperideal of , we get which implies . This means which is a contradiction. Thus is a unit element of . Hence we conclude that is local, by Lemma 1.

Theorem 3. Let be a 1-absorbing prime hyperideal of . If for nonunit elements and all proper hyperideal of , , then or .

Proof. Let for nonunit elements and all proper hyperideals of such that and . Therefore there exists an element but . Then we have such that and . This is a contradiction, since is 1-absorbing hyperideal of .

Theorem 4. Let be a -primary hyperideal of such that for every , . Then hyperideal of is 1-absorbing prime.

Proof. Let be a -primary hyperideal of such that for every , . Let for , but neither nor . Therefore we have , because hyperideal of is -primary hyperideal. Thus we get , a contradiction. Then we have either or which means hyperideal or is 1-absorbing prime.

Theorem 5. Let be a hyperideal of . If is a 1-absorbing hyperideal of , then is a 1-absorbing hyperideal of .

Proof. Suppose that for , . ThenIt is clear thatSince is a 1-absorbing hyperideal of , then we haveIt follows that either or . Therefore is a 1-absorbing hyperideal of .
Let be a multiplicative hyperring. We define the relation as follows:
if and only if where is a finite sum of finite products of elements of R, i.e.We denote the transitive closure of by . The relation is the smallest equivalence relation on a multiplicative hyperring such that the quotient , the set of all equivalence classes, is a fundamental ring. Let be the set of all finite sums of products of elements of R we can rewrite the definition of on as follows:We suppose that is the equivalence class containing . Then, both the sum and the product in are defined as follows: for all and for all Then is a ring, which is called a fundamental ring of [26].

Theorem 6. The hyperideal of is 1-absorbing hyperideal if and only if is a 1-absorbing ideal of .

Proof. Let for . Then, there exist such that and . Since , then . Since is a 1-absorbing hyperideal of , then or . Hencey or . Thus is a 1-absorbing ideal of .
Suppose that for . Then and . Since is a 1-absorbing ideal of , then we have or . It means or . Hence is a 1-absorbing hyperideal of . Tables 1 and 2.

Example 5. Let . Consider the multiplicative hyperring where “” and “” are defined on as follows:
It is easy to see that is a 1-absorbing prime hyperideal of . Then is a 1-absorbing ideal of .

Theorem 7. Let be a proper hyperideal of . Then the following statements are equivalent.(1) is a 1-absorbing prime hyperideal of .(2)If for some proper hyperideals of , then either or .

Proof. Let be a 1-absorbing prime hyperideal of and for some proper hyperideals of such that . Thus there exist nonunit elements and such that . Since and , we get by Lemma 3.7.
Let for some nonunit elements such that . Assume that , and . This means . Since , we conclude that and so .

#### 4. 1-Absorbing Primary Hyperideals

Definition 13. Let be a proper hyperideal of . is called a 1-absorbing primary hyperideal if for nonunit elements , , then either or .

Theorem 8. (1)Every primary hyperideal of is a 1-absorbing primary hyperideal.(2)Every 1-absorbing primary hyperideal of is a 2-absorbing primary hyperideal.

Example 6. In the multiplicative hyperring of integers with such that for any , hyperideal is a 1-absorbing primary hyperideal.

Example 7. Let be the ring of integers. We define the hyperoperation for all . Then is a multiplicative hyperring. The hyperideals and of are 1-absorbinh primary hyperideals.

Theorem 9. Let be a hyperideal of . If is 1-absorbing primary, then is a prime hyperideal of .

Proof. Let for some , . We suppose that are nonunit elements of . We assume that for , . Since , we have or , because is a 1-absorbing primary hyperideal of . Thus or which means is a prime hyperideal of .

Theorem 10. Let be a 1-absorbing primary strong C-hyperideal of such that is not a primary hyperideal. Then is local.

Proof. Let hyperideal of be 1-absorbing primary such that is not primary. Let for some nonunit elements , . By the assumption, we conclude that and . Let be a nonunit element of . Then we get . Since the hyperideal is 1-absorbing primary and , then . Let be a unit element of such that is a nonunit element of . It is clear that . Since is a 1-absorbing primary hyperideal of , we get , because . Since is a strong C-hyperideal of and , we conclude that . Since , we have which implies which is a contradiction. Hence is a unit element of . By Lemma 1, is local.
In the following theorem, we show that if is not a local multiplicative hyperring, then every 1-absorbing primary hyperideal is primary.

Theorem 11. Suppose that is not local and is a strong C-hyperideal of . Then is a 1-absorbing primary hyperideal of if and only if is a primary hyperideal of .

Proof. Let be a 1-absorbing primary hyperideal of and let for some nonunit elements such that . By Lemma 1, there exist a nonunit element and a unit element such that is a nonunit element of , because is not local. It is clear that . Since is a 1-absorbing primary hyperideal of and , then we get . Moreover, . Then , because is a 1-absorbing primary hyperideal of and . Since is a strong C-hyperideal of and , we have . Thus which implies . Consequently, is a primary hyperideal of .
It is clear.

Theorem 12. Let and be two multiplicative hyperrings. Let be a hyperideal of . The following statements are equivalent.(1) is a 1-absorbing primary hyperideal of .(2) is a primary hyperideal of .(3) for some primary hyperideal of .

Definition 14. A non-zero nonunit element of is called irreducible if for some , then is a unit of or is a unit of .

Definition 15. A non-zero element of is called prime if for some , then for some or for some .

Lemma 2. Let be a local strongly distributive multiplicative hyperring. If is a non-zero prime element of , then is an irreducible element of R.

Proof. Let is a non-zero prime element of such that for some , . We may assume that for some , . Now, we show that is a unit element of . Then we obtain . Hence . Let be a nonunit element of . Since is local, then is a unit element of which implies which is a contradiction. Thus is a unit element of . Consequently, is an irreducible element of .

Theorem 13. Let be a local strongly distributive multiplicative hyperring with maximal hyperideal . If element of is a non-zero prime element of with , then is a 1-absorbing primary hyperideal of that is not a primary hyperideal of .

Proof. Let for some nonunit elements , . Let . Then for all , and . Since for some , and for all , , then we have for some . Consequently, . Thus is a 1-absorbing primary hyperideal of . Now, since , then there exists but . It is clear that . By Lemma 2, is an irreducible element of , because is a non-zero prime element of . Then we conclude that . Since , we get . Thus is not a primary hyperideal of .

Theorem 14. Let is a 1-absorbing primary hyperideal of . Suppose that is not a primary hyperideal of . Then for some irreducible element and some nonunit element , . Moreover, if for some nonunit elements and , then element of is irreducible.

Proof. We suppose that is not a primary hyperideal of . Then for some nonunit elements and , . Assume that is not irreducible. Hence for some nonunit elements . Since is a 1-absorbing primary hyperideal of and and , then which implies which is a contradiction. Thus is irreducible.

Theorem 15. We suppose that is local with maximal hyperideal . Let be a prime hyperideal of with . Then the hyperideal of is a 1-absorbing primary hyperideal.

Proof. Let for some nonunit elements . Since , then . Let . Then and which means Thus . Consequently, hyperideal of is a 1-absorbing primary.

Theorem 16. Let the hyperideal be a 1-absorbing primary hyperideal of and let be a nonunit element of . Then is a primary hyperideal of .

Proof. Let for some nonunit elements . Let . Then . Since , then . Since is a 1-absorbing primary hyperideal of and , then . Therefore, the hyperideal is primary.

Definition 16. Let be a 1-absorbing primary hyperideal of . By Theorem 9, is a prime hyperideal of . Thus is called -1-absorbing primary hyperideal of .

Theorem 17. Let be -1-absorbing primary hyperideals of . Then is a -1-absorbing primary hyperideal of .

Proof. Put . It is clear that . Let for some nonunit elements , such that . Then there exist such that . Since is a -1-absorbing primary hyperideal of , then which means . Consequently, is a -1-absorbing primary hyperideal of .

Theorem 18. Let and be multiplicative hyperrings and be a good homomorphism such that if is a local multiplicative hyperring, then is a nonunit of for every nonunit . Then the following statements hold:(1)If is a 1- absorbing primary hyperideal of , then is a 1- absorbing primary hyperideal of .(2)If is an epimorphism and is a 1- absorbing primary hyperideal of containing , then is a 1- absorbing primary hyperideal of .

Proof. (1) Let be a local multiplicative hyperring. Let for some nonunit elements , . Thus we have . Since is a 1- absorbing primary hyperideal of , we conclude that which implies or which implies . Therefore, is a 1-absorbing primary hyperideal of . Now let be a 1- absorbing primary hyperideal of such that is not a local multiplicative hyperring. By Theorem 10, the hyperideal of is primary. Then we conclude that is a primary hyperideal of . Thus, is a 1-absorbing primary hyperideal of .
(2) Let for some nonunit elements , . Since is an epimorphism, there exist such that and so . Now take any . Then we get and so for some . This implies that , that is, and so . Since is a -hyperideal of , then we conclude that . Since is a 1-absorbing primary hyperideal of then which implies or which implies . Consequently, is a 1-absorbing primary hyperideal of .

Corollary 1. Let be two proper hyperideals of with such that if is local, then is a nonunit of for every nonunit . Then is a 1-absorbing primary hyperideal of if and only if is a 1-absorbing primary hyperideal of .

Proof. Define by . clearly, is a good epimorphism. Since and is a 1-absorbing primary hyperideal of , then the claim follows from Theorem 27 (2). Now let be a 1-absorbing primary hyperideal of . Then we conclude that is a 1-absorbing primary hyperideal of by Theorem 27 (1).

Theorem 19. Let the hyperideal of be 1-absorbing. If is a proper hyperideal of such that for some nonunit elements , then or .

Proof. Let such that neither nor . Thus there exists an element such that . It is clear that but and which is a contradiction, because is a 1-absorbing hyperideal of .
Next, it is proved that a proper hyperideal of is 1-absorbing primary if and only if for any proper hyperideals of implies that either or.

Theorem 20. Let I be a proper hyperideal of . Then is a 1-absorbing primary hyperideal of if and only if for any proper hyperideals of with implies that either or .

Proof. Let the hyperideal of be 1-absorbing primary. Suppose that for some proper hyperideals of , such that . This means there exist nonunit elements and but . Thus we get . Since is a 1-absorbing primary hyperideal of , then by Theorem 19.
Let for some nonunit elements , but . We have . Hence . By the assumption, we get which means . Thus the hyperideal of R is 1-absorbing.

#### 5. Strongly 1-Absorbing Primary Hyperideals

Definition 17. Let be a proper hyperideal of . is called a strongly 1-absorbing primary hyperideal if for nonunit elements , , then or .
It is clear that if is a strongly 1-absorbing primary hyperideal of then is a 1-absorbing primary hyperideal. Tables 3 and 4.

Example 8. We consider the ring that for each , and are the remainder of and , respectively, which “” and “” are ordinary addition and multiplication, and . which is a multiplicative hyperring. In the hyperring, hyperideal is a strongly 1-absorbing primary hyperideal of .

Theorem 21. If the hyperideals and of are strongly 1-absorbing primary, then is a strongly 1-absorbing primary hyperideal of .

Theorem 22. Let be a proper hyperideal of . Then, is a strongly 1-absorbing primary hyperideal of if and only if is 1-absorbing primary and , or is local with maximal hyperideal such that .

Proof. It is clear that if is a strongly 1-absorbing primary hyperideal of , then is a 1-absorbing primary hyperideal. Now, let . Suppose that for some , . We get for each . Since hyperideal is strongly 1-absorbing primary, we obtain which implies which means which is a contradiction. Thus for each nonunit elements , . Let the hyperideal of be maximal. Then which implies which means . Therefore for each maximal hyperideal of , . Consequently, is local.
Let be 1-absorbing primary and . Clearly, the hyperideal is strongly 1-absorbing primary. Let is local with the maximal hyperideal such that . Then we have for each nonunit elements . Then the hyperideal of is strongly 1-absorbing primary.
In view of Theorem 22, we have the following results.

Corollary 2. Let the hyperideal of be prime. Then, is strongly 1-absorbing primary if and only if or is local with maximal hyperideal .

Corollary 3. Let is local with maximal hyperideal and let be a prime hyperideal of . Then hyperideal of is strongly 1-absorbing primary if and only if or .

Proof. By Theorem 8 in [27], the hyperideal of is 1-absorbing primary. By Theorem 22, the hyperideal of is strongly 1-absorbing primary if and only if or and which implies or .

Theorem 23. There exists a strongly 1-absorbing primary hyperideal of if and only if is a prime hyperideal or is local.

Proof. Let be a strongly 1-absorbing primary hyperideal of . Suppose that is not local. Then , by Theorem 22. Since is a strongly 1-absorbing primary hyperideal of , then is a prime hyperideal, by Theorem 9. Let is local with maximal hyperideal . Then is a strongly 1-absorbing primary by Corollary 2. On the other hand, if is a prime hyperideal or is local, then by Corollary 2, is a strongly 1-absorbing primary of .

Corollary 4. If and are two multiplicative hyperrings. Then, has no strongly 1-absorbing primary hyperideal.

Proof. Since is not a prime hyperideal in and is not local, then has no strongly 1-absorbing primary hyperideal by Theorem 23.

Theorem 24. The hyperideal of is strongly 1-absorbing primary if and only if whenever for any proper hyperideals of with implies that either or .

Proof. Let for some proper hyperideals of , but . This means there exist and such that . It is clear that for each , . Since is a strongly 1-absorbing primary hyperideal of and , we get which implies . Suppose that for some nonunit elements but . It is clear that . Since , we have which implies . Thus, the hyperideal of is strongly 1-absorbing primary.

Theorem 25. The hyperideal is the only strongly 1-absorbing primary hyperideal of if and only if is a hyperfield or a non local integral hyperdomain.

Proof.