Abstract
Multiplicative hyperrings are an important class of algebraic hyperstructures which generalize rings further to allow multiple output values for the multiplication operation. Let be a commutative multiplicative hyperring. A proper hyperideal of is called 2-prime if for some , then, or . The 2-prime hyperideals are a generalization of prime hyperideals. In this paper, we aim to study 2-prime hyperideals and give some results. Moreover, we investigate -2-primary hyperideals which are an expansion of 2-prime hyperideals.
1. Introduction
The theory of algebraic hyperstructures was first introduced by Marty [1]. He defined the hypergroups as a generalization of groups in 1934. Since then, algebraic hyperstructures have been investigated by many researchers with numerous applications in both pure and applied sciences [2–9]. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Similar to hypergroups, hyperrings are algebraic structures more general than rings, substituting both or only one of the binary operations of addition and multiplication by hyperoperations. The hyperrings were introduced and studied by many authors [10–13]. Krasner introduced a type of the hyperring where addition is a hyperoperation and multiplication is an ordinary binary operation. Such a hyperring is called a Krasner hyperring [14]. A well-known type of a hyperring, called the multiplicative hyperring. The hyperring was introduced by Rota in 1982 which the multiplication is a hyperoperation, while the addition is an operation [15]. There exists a general type of hyperrings that both the addition and multiplication are hyperoperations [16]. Ameri and Kordi have studied Von Neumann regularity in multiplicative hyperrings [17]. Moreover, they introduced the concept of clean multiplicative hyperrings and studied some topological concepts to realize clean elements of a multiplicative hyperring by clopen subsets of its Zariski topology [18]. The notions such as (weak) zero divisor, (weak) nilpotent and unit in an arbitrary commutative hyperrings were introduced in the study by Ameri and Norouzi [19]. Some equivalence relations—called fundamental relations—play important roles in the theory of algebraic hyperstructures. The fundamental relations are one of the most important and interesting concepts in algebraic hyperstructures that ordinary algebraic structures are derived from algebraic hyperstructures by them. For more details about hyperrings and fundamental relations on hyperrings, see [1, 16, 20–23]. Prime ideals and primary ideals are two of the most important structures in commutative algebra. The notion of primeness of hyperideal in a multiplicative hyperring was conceptualized by Procesi and Rota [24]. Dasgupta extended the prime and primary hyperideals in multiplicative hyperrings [25]. Beddani and Messirdi [26] introduced a generalization of prime ideals called 2-prime ideals, and this idea is further generalized by Ulucak and Çelikel [27]. In [28], Dongsheng defined a new notion which is called -primary ideals in commutative rings. In [29], Ulucak introduced the concepts -primary and 2-absorbing -primary hyperideal over multiplicative hyperrings. In [30], we investigate -primary hyperideals in a Krasner -hyperring which unify prime hyperideals and primary hyperideals.
In this paper, we consider the class of multiplicative hyperring as a hyperstructure , where is an abelian group, is a semihypergroup, and the hyperoperation “” is distributive with respect to the operation “.” In this paper, we introduce and study the notion of 2-prime hyperideals of multiplicative hyperrings which are a generalization of prime hyperideals. Several properties of them are provided. Moreover, we investigate -2-primary hyperideals which are an expansion of 2-prime hyperideals. An earlier version of the manuscript has been presented as a preprint in https://www.researchgate.net [31].
2. Preliminaries
In this section, we give some definitions and results which we need to develop our paper.
A hyperoperation “” on a nonempty set is a mapping of into the family of all nonempty subsets of . Assume that “” is a hyperoperation on . Then, is called hypergroupoid. The hyperoperation on can be extended to subsets of as follows. Let be subsets of and , then,
A hypergroupoid is called a semihypergroup if for all , , which is associative. A semihypergroup is said to be a hypergroup if for all . A nonempty subset of a semihypergroup is called a subhypergroup if for all we have . A commutative hypergroup is canonical if(i)there exists with , for every ;(ii)for every there exists a unique with ; and(iii) implies .
A nonempty set with two hyperoperations “” and “” is called a hyperring if is a canonical hypergroup, is a semihypergroup with for all and the hyperoperation “” is distributive with respect to , i.e., and for all .
Definition 1. A 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 ; and(iii)for all , we have .If in (ii) the equality holds, then, we say that the multiplicative hyperring is strongly distributive. A nonempty subset of a multiplicative hyperring is a hyperideal,(i)If , then ;(ii)If and , then .Let be the ring of integers. Corresponding to every subset such that , there exists a multiplicative hyperring with and for any , .
Definition 2 (see [25]). A proper hyperideal of a multiplicative hyperring 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 the multiplicative hyperring does not have any prime hyperideal containing , we define .
Definition 3 (see [32]). A proper hyperideal of a multiplicative hyperring is maximal in if for any hyperideal of with , then, . Also, we say that is a local multiplicative hyperring, if it has just one maximal hyperideal.
Let be the class of all finite products of elements of , i.e., . A hyperideal of is said to be a -hyperideal of if, for any implies . Let be a hyperideal of . Then, where . The equality holds when is a -hyperideal of ([25], Proposition 1). In this paper, we assume that all hyperideals are -hyperideal.
Definition 4 (see [25]). A nonzero proper hyperideal of a multiplicative hyperring is called a primary hyperideal if for implies that or . Since is a prime hyperideal of by Proposition 3.6 in [25], is referred to as a -primary hyperideal of .
Definition 5 (see [32]). Let be a commutative multiplicative hyperring and be an identity (i. e., for all , ). An element is called unit, if there exists , such that .
Definition 6. A hyperring is called an integral hyperdomain, if for all , implies that or .
Definition 7 (see [32]). Let be a multiplicative hyperring. The element is an idempotent if .
Definition 8 (see [19]). An element is said to be zero divisor if there exists such that .
Definition 9. Let and be two multiplicative hyperrings. A mapping from into is said to be a good homomorphism if for all , , and .
Definition 10 (see [33]). A function is called a hyperideal expansion of if it assigns to each hyperideal of a hyperideal such that it has the following properties:(1).(2)If for any hyperideals of , then, .For example, consider the hyperideal expansions , , , and of defined with , , (for some hyperideal of ), and (for some hyperideal of ) for all hyperideals of , respectively. Also, let be a hyperideal expansion of and two hyperideals of such that . Let be defined by . Then, is a hyperideal expansion of .
Definition 11. (See [29]) Let be a hyperideal expansion of . A hyperideal of is called a -primary hyperideal if and imply either or .
Definition 12. (See [29]) Let be a good hyperring homomorphism, and hyperideal expansions of and , respectively. Then, is called a -homomorphism if for each hyperideal of .
Moreover, if is a -epimorphism and is a hyperideal of with , then, .
3. 2-Prime Hyperideals
Definition 13. Let be a proper hyperideal of a multiplicative hyperring . We say that is 2-prime if for all , implies or .
Example 1. Let be the ring of integers. In the multiplicative hyperring with and the hyperoperation for , the principal hyperideal is 2-prime.
Example 2. Consider the ring that for each , , and are the remainder of and , respectively, which and are ordinary addition and multiplication, and (see Tables 1 and 2)
is a multiplicative hyperring. In the hyperring, hyperideal is a 2-prime hyperideal of .
Note that an intersection of the 2-prime hyperideals may not be a 2-prime hyperideal of .
Example 3. Consider the ring of integers . We define for all . Then, is a multiplicative hyperring. The hyperideals and are 2-prime, but since and .
Theorem 1. Let be a hyperideal of a multiplicative hyperring . Then,(1)If is a 2-prime hyperideal, then is a prime hyperideal.(2)If is a prime hyperideal of , then is a 2-prime hyperideal of .(3)Let and be two multiplicative hyperrings such that is a good homomorphism. If is a 2-prime hyperideal of , then is a 2-prime hyperideals of .(4)Let and be two multiplicative hyperrings such that is a good epimorphism. If is a 2-prime hyperideal of with , then, is a 2-prime hyperideal of .(5)If is a 2-prime hyperideal and are two subsets of with , then, or .
Proof. (1)Let the hyperideal be 2-prime. If for , then, for some positive integer , . Since hyperideal is 2-prime, we have or . Thus, we get or which means hyperideal is prime.(2)Since , then, we are done.(3)Let for . Hence, . Since the hyperideal is 2-prime, then, or . Thus, or , i.e., the hyperideal is 2-prime.(4)Let for some 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 2-prime hyperideal of then, which implies or which implies . Consequently, is a 2-prime hyperideal of .(5)Let or . Hence, there exist and with and . Since hyperideal is 2-prime, then, which is a contradiction.Let be a 2-prime hyperideal of . Since is a prime hyperideal of by Theorem 1. (1), is referred to as a -2-prime hyperideal of .
Theorem 2. Let be a -2-prime hyperideal of a multiplicative hyperring . Then, for all is a -2-prime hyperideal of .
Proof. Let for . This means that . Since the hyperideal is -2-prime and , then, which means . Therefore, we have . Since , then, . Since , then, we have . Assume that for , such that . We have . Since the hyperideal is -2-prime and , then, . This means . Consequently, is a -2-prime hyperideal of .
In reference [17], recall that an element is called regular if there exists such that . So, we can define that is regular multiplicative hyperring, if all of elements in are regular elements.
Theorem 3. Let be a regular multiplicative hyperring and be a 2-prime hyperideal of . Then, is prime.
Proof. Let be a 2-prime hyperideal of . Suppose that with for some . Since is regular, then, there exists such that . Let . Then, which is a contradiction. Therefore, . Since is a 2-prime hyperideal of , then, . This means . Consequently, the hyperideal is prime.
Recall that a proper hyperideal of a multiplicative hyperring is called semiprime if whenever for some and , then, . Note that if is a semiprime hyperideal of such that for some hyperideal of and , then, (Proposition 2 in [34]). Every prime hyperideal is a semiprime hyperideal, but, the converse is not true in general. For example, the hyperideal of is semiprime, but it is not prime (see Example 2.3 in [34]).
Theorem 4. Let be a semiprime hyperideal of a multiplicative hyperring and be a hyperideal expansion of . Then, is a 2-prime hyperideal of if and only if is a prime hyperideal of .
Definition 14. A proper hyperideal of a multiplicative hyperring is called semiprimary if for all such that , then, either or lies in .
Example 4. Let be the ring of integers. In the multiplicative hyperring with and hyperoperation for , the hyperideal is semiprimary.
Theorem 5. Every 2-prime hyperideal of is a semiprimary hyperideal of .
Theorem 6. Let be a multiplicative hyperring of integers and be a positive integer such that each element of and are coprime and . Then, is a 2-prime hyperideal of if and only if for some positive integer and an irreducible element of or .
Proof. Let be a 2-prime hyperideal of . Assume that is not an irreducible element. Then, is a representation of a as a product of distinct prime integers such that is a positive integer for . Put and by Proposition 4.13 in [25]. For every , which implies . Since is 2-prime, then, we have or . If , then, for all we have . Thus, we obtain for some . Since each element of and is coprime, then, for some , we get which is contradiction. If then for every , which implies for which means for some which is a contradiction.
Let for some positive integer and an irreducible element . Let for some . Then, for , which implies and for some with . If and , then, which is a contradiction. Therefore, for some or which implies or . Consequently, is a 2-prime hyperideal of .
Let be a multiplicative hyperring. Then, we call as the set of all hypermatixes of . Also, for all if and only if [32].
Theorem 7. Let be a multiplicative hyperring with scalar identity 1 and be a hyperideal of . If is a 2-prime hyperideal of , then, is an 2-prime hyperideal of .
Proof. Suppose that for , . Then,It is clear thatsince is a 2-prime hyperideal of , then,which means orwhich means . Therefore, is a 2-prime hyperideal of .
Let be a hyperring. We define the relation as follows:
if and only if where is a finite sum of finite products of elements of , i.e., such that . 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 we can rewrite the definition of on as follows:
if and only if there exist with and such that for . 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 (see also [9]).
Theorem 8. Let be a multiplicative hyperring with scalar identity 1. Then, a hyperideal of is 2-prime if and only if is a 2-prime ideal of .
Proof. Let for . Thus, there exist such that and . So, , then, . Since is 2-prime hyperideal, then, or . Hence, or . Thus, is a 2-prime ideal of . Suppose that for , then, and . Since is a 2-prime ideal of , then, we have or . It means that or . Hence, is a 2-prime hyperideal of .
Lemma 1. Let be a local multiplicative hyperring with maximal hyperideal and be a prime hyperideal of . Then, is a 2-prime hyperideal of . Moreover, if is a prime hyperideal of , then, .
Proof. Let for some . This means or . Suppose that . Clearly, is not unit. Then, we have which implies . Thus, the hyperideal is 2-prime.
For the second assertion, assume that the hyperideal is prime. Suppose that . Since , then, which implies because is a prime hyperideal of . This means . Since , then, we have .
Theorem 9. Let be a local multiplicative hyperring with maximal hyperideal . Then, every 2-prime hyperideal of is prime if and only if for each minimal prime hyperideal over an arbitrary 2-prime hyperideal , . Furthermore, if every 2-prime hyperideal of is prime, then, is an idempotent hyperideal.
Proof. Assume that every 2-prime hyperideal of is prime. Since the hyperideal is 2-prime, then, is prime. Then, is a 2-prime hyperideal of , by Lemma 1. This means is a prime hyperideal of . Then, we conclude that , by Lemma 1.
Suppose that is a 2-prime hyperideal of . Then, we have , by Theorem 1 (1). By the assumption, we get . This implies that is a prime hyperideal of .
Theorem 10. Let every 2-prime hyperideal of a multiplicative hyperring be prime and be an arbitrary prime hyperideal of . Then, .
Proof. Assume that every 2-prime hyperideal of is prime and is an arbitrary prime hyperideal of . Then, the hyperideal is 2-prime, by Theorem 1 (2). Since very 2-prime hyperideal of is prime, then, the hyperideal is prime. It is easy to see that is equal to .
Definition 15. Let be a hyperideal of a multiplicative hyperring . We say that a 2-prime hyperideal is minimal over if there is no a 2-prime hyperideal of with . Note that 2- denotes the set of minimal 2-prime hyperideals over .
Theorem 11. Let be a local multiplicative hyperring with maximal hyperideal and be a prime hyperideal of . Let for each 2-prime hyperideal of , . Then, the followings are equivalent:(1)If for each hyperideal , then, .(2)If for each hyperideal , then, .
Proof. (1) (2) Suppose that for some . First, we show that the hyperideal is minimal over . Assume that we have for some prime hyperideal of . By the assumption, we get . Let . Then, we obtain which implies . This means , since is a prime hyperideal of . Thus, we conclude that the hyperideal is minimal over . Now, since and , then, we get .
(2) (1) Let such that . We get since the hyperideal is prime. Clearly, . By Lemma 1, is a 2-prime hyperideal of . Since and , then, .
Corollary 1. Let be a local multiplicative hyperring with maximal hyperideal and be a prime hyperideal of . Let for each -2-prime hyperideal , . Then, for each hyperideal with , if and only if every 2-prime hyperideal of is prime.
Proof. The claim follows by Theorems 9 and 11.
4. Expansion of 2-Prime Hyperideals
Definition 16. Let be a hyperideal expansion of a multiplicative hyperring . A proper hyperideal of is called -2-primary if for , implies either or .
Example 5. Suppose that is the ring of integers. For all , we define the hyperoperation . Then, is a multiplicative hyperring. Consider the hyperideal expansion by . Since , then, we conclude that is a -2-primary hyperideal of .
Clearly, every prime hyperideal of a multiplicative hyperring is a -2-primary hyperideal but its inverse is not true in general.
Example 6. Let be the ring of integers, be the set of all even integers of and be the set of all positive even integers of . In the multiplicative hyperring (see Example 3.5 in [25]), the hyperideal is -2-primary but is not prime.
Example 7. In the multiplicative hyperring with trivial hyperoperation, i.e., for , the hyperideal is -2-primary.
We start the section with the following trivial result, and hence, we omit its proof.
Theorem 12. Let be a proper of . Then,(1) is a -2-primary hyperideal if and only if is a 2-prime hyperideal.(2)If is a primary hyperideal, then, is a -2-primary hyperideal.(3)If is a 2-prime hyperideal, then, is a -2-primary hyperideal for every hyperideal expansion of .(4)If is a -primary hyperideal, then, is a -2-primary hyperideal for every hyperideal expansion of .(5)If is a -2-primary hyperideal of such that for some hyperideal expansion of , then, is a -2-primary hyperideal of .
Theorem 13. Let be a hyperideal of a multiplicative hyperring and a hypoerideal expansion of . If is a -2-primary hyperideal of , then, for some idempotent element , is a -2-primary hyperideal of .
Proof. Let such that for some . This means that but . Since is a -2-primary hyperideal of , we get . Thus, is a -2-primary hyperideal of .
Theorem 14. Let be a hyperideal of a multiplicative hyperring such that for each , and a hyperideal expansion of . If the hyperideal is irreducible, then, is a -2-primary hyperideal of .
Proof. We suppose that is not a -2-primary hyperideal of and look for a contradiction. This means that there exist such that but and . Thus, we get and . Let . Then, there are and such that for some and we have . So, we get . Also, we have . Since , then, . This implies that . Since is a -hyperideal of , then, . Hence, which implies . Therefore, which means , by the assumption. So, . Thus, we have . Then, . Since , then, we obtain . This is a contradiction since is irreducible. Consequently, the hyperideal of is -2-primary.
Theorem 15. Let be a hyperideal expansion of a multiplicative hyperring . Let and be semiprime hyperideals of . Then, is a -2-primary hyperideal of if and only if is -primary.
Theorem 16. Let be a hyperideal of a multiplicative hyperring and a hyperideal expansion of such that . If is a -2-primary hyperideal of , then, is a -primary hyperideal of .
Proof. Let for some such that . This means that we have for some . Clearly, . Since is a -2-primary hyperideal of , then, which implies . This means that is a -primary hyperideal of .
Theorem 17. Let be a hyperideal of a multiplicative hyperring and a hyperideal expansion of such that . If the hyperideal is primary, then, is a -2-primary hyperideal of .
Proof. Let for some . We have or . Thus, or . Consequently, is a -2-primary hyperideal of .
Recall from [29] that a hyperideal expansion of has the property of intersection preserving, if it satisfies for any hyperideals and of .
Theorem 18. Let has the property of intersection preserving. If is a -2-primary hyperideal of with for all . Then, is a -2-primary hyperideal of .
Proof. Let with and . Then, there exists some such that . Since is a -2-primary hyperideal of , then, . Therefore, . Thus, we conclude that is a -2-primary hyperideal of .
Theorem 19. If is a directed set of -2-primary hyperideals of a multiplicative hyperring , then, is a -2-primary hyperideal of .
Proof. Let for some . Assume that . This implies that there exists such that . Since is a -2-primary hyperideal of , then, . Thus, is a -2-primary hyperideal of .
Theorem 20. Let be a regular multiplicative hyperring and be hyperideal expansion of . If is a -2-primary hyperideal of , then, is -primary.
Proof. Let is a -2-primary hyperideal of . Assume that and such that . Since is regular, then, there exists such that . Let . Then, which is a contradiction. So, . Since is a -2-primary hyperideal of , then, . Since is regular, then, there exists such that . Therefore, the hyperideal is -primary.
Theorem 21. Let and be hyperideal expansions of and , respectively, and a -homomorphism. Then,(1)If is a -2-primary hyperideal of , then, is a -2-primary hyperideal of .(2)Let be a hyperideal of and an epimorphism with . Then, is a -2-primary hyperideal of if and only if is a -2-primary hyperideal of .
Proof. (1) Let for some . Then, we have . Since is a -2-primary hyperideal of , we get or which implies or . Then, or because is a -homomorphism. Consequently, is a -2-primary hyperideal of . (2) It is quite clear from (1).
Let for some . Then, for some we have and . So, . Now, take any . Then, and so there exists such that . This means that , that is, and then, . Since is a -hyperideal of , then, we get . Since is a 2-primary hyperideal of , then, we obtain or . This implies that or . Thus, is a -2-primary hyperideal of .
Corollary 2. Let and be two hyperideals of a multiplicative hyperring such that . Then, is a -primary hyperideal of if and only if is a -primary hyperideal of .
Proof. The claim is verified from Theorem 21.
Corollary 3. Let be a hyperideal of a multiplicative hyperring and a subhyperring of with . If is a -2-primary hyperideal of , then, is a -2-primary hyperideal of .
Theorem 22. Let be a hyperideal expansion of . Then, the followings are equivalent:(1)Every proper principal hyperideal of is -2-primary.(2)Every proper hyperideal of is -2-primary.
Proof. (1) (2) Assume that is a proper hyperideal of such that for some . Take any . Clearly, . This implies that . Since is a -2-primary hyperideal of , then, we get or . Therefore, is a -2-primary hyperideal of . It is clear.
Let and be two multiplicative hyperrings with nonzero identity [29]. Recall is a multiplicative hyperring with the operation and the hyperoperation are defined respectively as and .Assume that and are hyperideal expansions of and , respectively. If is a function of hyperideals of with for every hyperideals and of and , respectively, then, is a hyperideal expansion of .
Theorem 23. Let and be two multiplicative hyperrings with nonzero identity such that and be hyperideal expansions of and , respectively. Let be a hyperideal of . Then, the hyperideal of is -2-primary if and only if is a -2-primary of .
Proof. Let for some . This means . Since is a -2-primary hyperideal of , then, we get or . This implies that or .
Assume on the contrary that is not a -2-primary hyperideal of . So, with implies that and . It is clear that . Since is a -2-primary hyperideal of , then, we have or . Hence, we get or which is a contradiction. Consequently, is a -2-primary hyperideal of .
Theorem 24. Let and be two multiplicative hyperrings with nonzero identity such that and be hyperideal expansions of and , respectively. Let and be some hyperideals of and , respectively. Then, the following statements are equivalent:(1) is a -2-primary hyperideal of .(2) and is a -2-primary hyperideal of or and is a -2-primary hyperideal of .
Proof. (1) (2) Assume that . Then, is a -2-primary hyperideal of , by Theorem 23.
(2) (1) This can be proved, by using Theorem 23.
Example 8. Suppose that is the ring of integers. Then, is a multiplicative hyperring with a hyperoperation . Also, is a multiplicative hyperring with a hyperoperation . Note that is a multiplicative hyperring with a hyperoperation . Clearly, and are two -2-primary of and , respectively. Since , but and , then, is not a -2-primary hyperideal of .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.