-Hyperideals and Generalizations of -Hyperideals in Krasner Hyperrings
This paper deals with Krasner hyperrings as an important class of algebraic hyperstructures. We investigate some properties of -hyperideals in commutative Krasner hyperrings. Some properties of -hyperideals are also studied. The relation between prime hyperideals and -hyperideals is investigated. We show that the image and the inverse image of an -hyperideal are also an -hyperideal. We also introduce a generalization of -hyperideals, and we prove some properties of them.
Prime ideals and primary ideals play a significant role in commutative ring theory. Numerous generalizations of prime ideals have been studied by different researchers. The concepts of -prime and -primary ideals are examined in [1, 2].
French mathematician Marty  initiated the study of hyperstructures, as an expansion of classical algebraic structures, in special, of hypergroups, at the 8th Congress of Scandinavian Mathematicians in 1934. Subsequently, numerous articles and various books have been published about this issue. This theory has been studied by many researchers [4–17].
Hyperrings generalize rings, and they arise naturally in several subfields in algebra, including quadratic form theory, number theory, orderings and ordered algebraic structures, tropical geometry, and multiplicative subgroups of fields. Hyperrings are of different types. If is a binary operation and the multiplication is a hyperoperation, then the hyperring is called multiplicative hyperring . Krasner hyperrings are an important class of algebraic hyperstructures that generalize rings further to allow multiple output values for the addition operation . In fact, Krasner introduced the concepts of hyperrings and hyperfields in 1956 . In 2006, the same notions called “multirings” and “multifields” were introduced independently by Marshall , with the only difference between hyperrings and multirings to be that hyperrings have the strong distributive property whereas multirings have the weak distributive property. Later on, after Krasner, Stratigopoulos  and Mittas [13, 14, 22] have initiated the general study of these algebraic hyperstructures. There are several well-known authors that have made an important contribution to the study of hyperrings and hyperfields later on and nowadays such as Massouros [23–25], Nakassis , Vougiouklis , Spartalis [28, 29], G. Pinotsis, Y. Kemprasit, M. Stefanescu, V. Leoreanu, R. Ameri, I. Cristea, and many others. The study of Krasner hyperrings has been recently one of the mainstream objective of the researchers [9, 30–32].
In 2015, Mohamadian  investigated some properties of -ideals in commutative rings. While the research of Erbay et al.  focuses on -ideals in commutative semigroups, Koc and Tekir in  considered a generalization of this notion to modules. In , Ugurlu studied generalizations of -ideals. The concept of -hyperideals in commutative hyperrings was briefly mentioned in , as a generalization of -ideals in commutative rings.
In this paper, our aim is to extend the notion of -ideals to -hyperideals in Krasner hyperrings and generalize them. Some properties of -hyperideals in the commutative Krasner hyperrings are obtained. Some properties of -hyperideals are also studied. The relation between prime hyperideals and -hyperideals is investigated. It is shown that if is a minimal prime hyperideal, then it is also an -hyperideal. Furthermore, maximal hyperideals and -hyperideals are compared. We show that the image and the inverse image of an -hyperideal are also an -hyperideal. Moreover, a generalization of -hyperideals is introduced. -prime and -primary hyperideals, -hyperideals, -hyperideals, pure hyperideals, and von Neumann regular hyperideals in are introduced and studied. Our study serves as a continuation of the study in more depth of the results of published papers reflected in the references on Krasner hyperrings, introducing and investigating these new classes of hyperideals.
Let be a commutative Krasner hyperring, be a proper hyperideal of , and be a function. denotes the hyperideals of . is called -hyperideal (resp., -hyperideal) if with implies that (resp., ), for . Some properties of them are provided. As a result, we obtain that the union of directed collection of ascending chain -hyperideals of is also -hyperideal of , when preserves the order.
For convenience, let us first give the definitions of prime ideal, primary ideal, and -ideal in a commutative ring.
Definition 1. (see ). Let be a commutative ring. A reduction of ideals is a function that leads any ideal of to other ideal of such that the following statements hold:(i)For all ideals of , (ii)If where and are ideals of , then
Definition 3. (see ). Let be a commutative ring. A proper ideal of is called an -ideal (resp., -ideal), if with implies that (resp., , for any ), for each .
Definition 4. (see ). Let be a commutative ring. A proper ideal of is called a -ideal, if with implies that for . Furthermore, is called a -ideal, if with implies that for .
In the following, we recall some notions regarding hyperstructure.
Let be a nonempty set and denotes the family of all nonempty subsets of . A mapping is called a binary hyperoperation on . The couple is called a hypergroupoid . In the above definition, if and are two nonempty subsets of and , then we defineA hypergroupoid is said to be a semihypergroup if for all , , which means thatWhen is a hypergroupoid, if there is such that , for every , then is called the identity element.
Semihypergroup is said to be a hypergroup if , . If is a hypergroup, is a subset of and , , then is called a subhypergroup of .
Let be a hypergroup. If , , then is called a commutative hypergroup.
Definition 5. (see ). A nonempty set along with the hyperoperation is called a canonical hypergroup if the following axioms hold:(i), for (ii), for (iii)There exists such that , for any (iv)For any , there exists a unique element , such that ( is called the opposite of and it is denoted by )(v) implies that and ; that is, is reversible
Definition 6. (see ). is called a Krasner hyperring if it satisfies the following conditions:(1) is a canonical hypergroup(2) is a semigroup having 0 as a bilaterally absorbing element, that is, , for all (3) and , for all
Lemma 1 (see ). A nonempty subset of a Krasner hyperring is a left (resp. right) hyperideal if and only if(i), for (ii), for
Definition 7. (see ). Let be a hyperideal of a hyperring . Then, , for some .
Let be a hyperring. For , we define . If (resp., ), is said to be regular (resp., zero divisor). We use the notion to denote the set of all regular elements (resp., zero divisors) . If is a hyperideal of and a subset of , then we denote . It is clear that .
Definition 8. (see ). A proper hyperideal of a commutative Krasner hyperring is called an -hyperideal (-hyperideal), if and implies that (, for some ), for any .
is the set of all minimal prime hyperideals which contain . is stated by .
Definition 9. (cf. ). A hyperring satisfies the following:(i)Property A: if any finitely generated hyperideal, has nonzero annihilator(ii)Annihilator condition: if for any finitely generated hyperideal of , there exists an element such that (iii)Strong annihilator condition (briefly s.a.c.): if for any finitely generated hyperideal of , there exists an element such that is a -hyperideal if , and imply that .
denotes the sum of all minimal hyperideals of . The socle of a reduced hyperring is called , where is the set of idempotents of .
Let be a subset of and such that . Then, is called von Neumann regular element. Therefore, if all of the elements are von Neumann regular, or is called von Neumann regular .
A two-sided hyperideal in a semihypergroup is called right pure hyperideal if, for each , there exists such that . Similarly, a hyperideal is called pure hyperideal in a Krasner hyperring, if for each , there exists such that .
A hyperring is called a reduced hyperring if there are no nilpotent elements in . If for , then .
Let be a homomorphism, be a hyperideal of , and be a hyperideal of . Then, the hyperideal is said to be the extension of , and it is denoted by . The hyperideal is called contraction of and denoted by . The mapping given by is a homomorphism. If is a hyperideal of , then is also a hyperideal of . If , then the hyperring is called quotient hyperring which is denoted by .
In this section, we examine some properties of -hyperideals in commutative hyperrings. We compare -hyperideals with prime and maximal hyperideals. Throughout the section, is a commutative Krasner hyperring.
Initially, let us give some examples for better understanding.
Example 2. Clearly, is a Krasner hyperring with the usual addition and multiplication, and is a proper hyperideal. and implies that and implies that and implies that Then is an -hyperideal of .
Example 3. (see ). Let and . Then, is a Krasner hyperring with the hyperaddition and multiplication defined by
Clearly, is an -hyperideal of .
Example 4. Let be a set with the hyperaddition and multiplication defined as follows:
Then, is a Krasner hyperring . It can be easily seen that are -hyperideals.
Our main results regarding -hyperideals are the followings.
Theorem 1. Let be a hyperring and be a hyperideal of . Then, the following statements are equivalent:(a) is an -hyperideal(b), for any (c), for any (d), where is a hyperideal of
Proof. Let be an -hyperideal and be a regular element. Suppose that . Then, and . Thus, , for . Since , and is an -hyperideal, . Hence, and .
Therefore, for every , , .
We know that for every . Let be regular, and . Hence, and . From (b), and This implies that , for . Since , then . Hence, . Thus, .
Let be the set of regular elements and be a natural homomorphism. We know that , for a hyperideal of . Suppose . Since , then , for . From (c), .
Let and . We have , and since is regular, then there exists which is the inverse of in . Thus, and . Hence, and so is an -hyperideal.
Corollary 1. The following statements hold:(a)The zero hyperideal is an -hyperideal(b)The intersection of -hyperideals is an -hyperideal(c)When is an -hyperideal, (d)Every -hyperideal is a -hyperideal(e)A prime hyperideal is an -hyperideal if and only if it consists all of zerodivisors. As a result, every minimal prime hyperideal is an -hyperideal(f)Let be an -hyperideal, and . Then, is an -hyperideal. Particularly, and are always -hyperideals.(g)Every minimal hyperideal of a reduced hyperring is an -hyperideal(h)Every pure hyperideal and every von Neumann regular hyperideal are -hyperideals(i)Suppose that satisfies the s.a.c. and is a hyperideal of is an -hyperideal if and only if for every hyperideal and of such that is finitely generated, whenever and , then (j)The sum of two -hyperideals may not be an -hyperideal
Proof. (b) Suppose that are -hyperideals of . Let and . Then, . Since every is -hyperideal, then we have . Hence, ., which means the intersection of -hyperideals is also -hyperideal. (c) Let be an -hyperideal of and . Then, there exists a regular element in . Now, let be identity element of . In this way, . Since is an -hyperideal, then . This is a contradiction to ’s being proper hyperideal. (d) If a prime hyperideal is an r-hyperideal, then it consists all of zerodivisors from . For the converse, suppose that is prime and . Let and . Since is prime, then or . Since we assume , then there is no regular element in and so . Then, we get . We conclude that is an r-hyperideal. (i) Assume that is an -hyperideal, is finitely generated hyperideal, and is a hyperideal of such that and . Then, there exists a such that and so . For , we have . Since is an -hyperideal, then . That’s why . Conversely, assume that for every hyperideal and of such that is finitely generated, whenever and , then . Let and . Then, and . Therefore, because of . Thus, , and we get the desired result. (j) To prove that the sum of two -hyperideals may not be an -hyperideal, we give the following example:
Example 5. Let be a group and , where 0 is an absorbing element under multiplication and are distinct orthogonal idempotents with ; for all ; for all If we define the hyperaddition on as follows: , for all , for all and Then, is a Krasner hyperring . Consider and . It can be easily seen that are -hyperideals, while is not an -hyperideal.
We omit the rest of proof since it is obvious.
Remark 1. Let and be hyperideals of . We know that and . If and are -hyperideals of , then by Theorem 1(d), and , for some hyperideals and in . It follows that(a) is an -hyperideal of if and only if (essentially )(b) is an -hyperideal of if and only if (essentially )
Lemma 2. Let be a hyperring and be a hyperideal. Then, the following statements hold:(a) is an -hyperideal if and only if for any , hyperideals of with and , then (b)Assume that is not an -hyperideal. There exist hyperideals and such that , , , and
Proof. (a)Let be an -hyperideal, and be hyperideals of such that and . Let and . Since and are hyperideals, then we can take and . By our assumption, . Since , then let we take . Then is an -hyperideal and so . Thus, . Conversely, assume that and . Let and . Then, . Let . So . At the same time, since , then .(b)Assume that is not an -hyperideal. Then, there exist and such that . We have . Let and . It follows . Since , then and . Since , then . Therefore, . Thus, there exist such that and . Hence, . Then, and .
Proposition 1. (a)Let be a hyperring and be a hyperideal of with . If and are -hyperideals of such that or , then .(b)Let be a hyperring and be hyperideals of with . If is -hyperideal of , then . In addition, is an -hyperideal.
Theorem 2. Let be prime hyperideals of , which are not comparable. If is an -hyperideal, then is an -hyperideal, for .
Proof. Let such that and . Let . Then, . We get that , since is an -hyperideal and . Thus, . We conclude that , since and ’s are prime. Hence, is an -hyperideal.
Let be a homomorphism. We investigate whether the image of an -hyperideal and the inverse image of an -hyperideal are -hyperideal.
Theorem 3. Let be a good epimorphism such that . If is an -hyperideal of , then is an -hyperideal of the hyperring .
Proof. Obviously, is hyperideal of . Let and , for . Since is onto, then there exist such that and . Then, , for some .
. Then, there exists such that . We have . Thus, .
Let . Then, there exists such that . So, . Since , then , and this is a contradiction. This implies that . Since is an -hyperideal, then . Therefore, .
Theorem 4. Let be a good monomorphism. If is an -hyperideal of , then is an -hyperideal of .
Proof. is a hyperideal . Let and . Then, . If , then there exists such that . This means that there exists a such that . Then, . This is a contradiction, so . Since is an -hyperideal, then and therefore .
Theorem 5. Let be a hyperring. The following statements are equivalent:(a) is a hyperdomain(b)The only -hyperideal of is the zero hyperideal(c), for each
Proof. Let us suppose that is hyperdomain and is a proper hyperideal of . Then, there exists . Since is a hyperdomain, then we have . This contradicts the hyperideal of being proper. Then, the zero hyperideal is the only -hyperideal. From Corollary 1 (f), is an -hyperideal. Assume that the zero hyperideal is the only -hyperideal of . Hence, . So , for every Let , . Thus,