# r-Hyperideals and Generalizations of r-Hyperideals in Krasner Hyperrings

Authors: Peng Xu; Melis Bolat; Elif Kaya; Serkan Onar; Bayram Ali Ersoy; Kostaq Hila
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper deals with Krasner hyperrings as an important class of algebraic hyperstructures. We investigate some properties ofr-hyperideals in commutative Krasner hyperrings. Some properties of pr-hyperideals are also studied. The relation between prime hyperideals and r-hyperideals is investigated. We show that the image and the inverse image of an r-hyperideal are also an r-hyperideal. We also introduce a generalization of r-hyperideals, and we prove some properties of them.


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## Body

## 1. Introduction

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 ofϕ-prime and ϕ-primary ideals are examined in [1, 2].French mathematician Marty [3] 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. Hyperringsℜ,+,⋅ are of different types. If + is a binary operation and the multiplication ⋅ is a hyperoperation, then the hyperring is called multiplicative hyperring [18]. Krasner hyperrings are an important class of algebraic hyperstructures that generalize rings further to allow multiple output values for the addition operation [12]. In fact, Krasner introduced the concepts of hyperrings and hyperfields in 1956 [19]. In 2006, the same notions called “multirings” and “multifields” were introduced independently by Marshall [20], 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 [21] 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 [26], Vougiouklis [27], 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 [33] investigated some properties of r-ideals in commutative rings. While the research of Erbay et al. [34] focuses on r-ideals in commutative semigroups, Koc and Tekir in [35] considered a generalization of this notion to modules. In [36], Ugurlu studied generalizations of r-ideals. The concept of r-hyperideals in commutative hyperrings was briefly mentioned in [37], as a generalization of r-ideals in commutative rings.In this paper, our aim is to extend the notion ofr-ideals to r-hyperideals in Krasner hyperrings and generalize them. Some properties of r-hyperideals in the commutative Krasner hyperrings are obtained. Some properties of pr-hyperideals are also studied. The relation between prime hyperideals and r-hyperideals is investigated. It is shown that if N is a minimal prime hyperideal, then it is also an r-hyperideal. Furthermore, maximal hyperideals and r-hyperideals are compared. We show that the image and the inverse image of an r-hyperideal are also an r-hyperideal. Moreover, a generalization of r-hyperideals is introduced. ϕ-prime and ϕ-primary hyperideals, ϕ−r-hyperideals, ϕ−pr-hyperideals, ϕ− pure hyperideals, and ϕ− von Neumann regular hyperideals in ℜ 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.Letℜ be a commutative Krasner hyperring, N be a proper hyperideal of ℜ, and ϕ:Idℜ⟶Idℜ∪∅ be a function. Idℜ denotes the hyperideals of ℜ. N is called ϕ−r-hyperideal (resp., ϕ−pr-hyperideal) if a⋅b∈N−ϕN with anna=0 implies that b∈N (resp., bn∈N), for a,b∈ℜ. Some properties of them are provided. As a result, we obtain that the union of directed collection of ascending chain ϕ−r-hyperideals of ℜ is also ϕ−r-hyperideal of ℜ, when ϕ preserves the order.


## 2. Preliminaries

For convenience, let us first give the definitions ofϕ− prime ideal, ϕ− primary ideal, and r-ideal in a commutative ring.Definition 1.
(see [38]). Let ℜ be a commutative ring. A reduction of ideals is a function ϕ that leads any ideal N of ℜ to other ideal ϕN of ℜ such that the following statements hold:(i)
For all idealsN of ℜ, ϕN⊆N(ii)
IfN⊆M where N and M are ideals of ℜ, then ϕN⊆ϕMDefinition 2.
Letℜ be a commutative ring. Let N be an ideal of ℜ. For a,b∈ℜ,(i)
N is said to be a ϕ− prime ideal, if ab∈N−ϕN, then a∈N or b∈N [1](ii)
N is said to be a ϕ− primary ideal, if ab∈N−ϕN, then a∈N or b∈N [2]Definition 3.
(see [33]). Let ℜ be a commutative ring. A proper ideal N of ℜ is called an r-ideal (resp., pr-ideal), if ab∈N with anna=0 implies that b∈N (resp., bn∈N, for any n∈ℕ), for each a,b∈ℜ.Definition 4.
(see [36]). Let ℜ be a commutative ring. A proper ideal N of ℜ is called a ϕ−r-ideal, if ab∈N−ϕN with anna=0 implies that b∈N for a,b∈ℜ. Furthermore, N is called a ϕ−pr-ideal, if ab∈N−ϕN with anna=0 implies that bn∈Nn∈ℕ for a,b∈ℜ.
In the following, we recall some notions regarding hyperstructure.
LetG be a nonempty set and P∗G denotes the family of all nonempty subsets of G. A mapping ∘:G×G⟶P∗G is called a binary hyperoperation on G. The couple G,° is called a hypergroupoid [3]. In the above definition, if A and B are two nonempty subsets of G and x∈G, then we define(1)A∘B=∪a∈A,b∈Ba∘b,A∘x=A∘x,andx∘B=x∘B.
A hypergroupoidH,∘ is said to be a semihypergroup if for all x,y,z∈H, x∘y∘z=x∘y∘z, which means that(2)∪u∈x∘yu∘z=∪v∈y∘zx∘v.
WhenG,∘ is a hypergroupoid, if there is e∈G such that x∈e°x∩x°e, for every x∈G, then e is called the identity element.
SemihypergroupG,∘ is said to be a hypergroup if x∘G=G∘x=G, ∀x∈G [3]. If G,∘ is a hypergroup, ∅≠H is a subset of G and x∘H=H∘x=H, ∀x∈H, then H,∘ is called a subhypergroup of G,∘.
LetG,∘ be a hypergroup. If x∘y=y∘x, ∀x,y∈G, then G,∘ is called a commutative hypergroup.Definition 5.
(see [14]). A nonempty set ℜ along with the hyperoperation + is called a canonical hypergroup if the following axioms hold:(i)
x+y+z=x+y+z, for x,y,z∈ℜ(ii)
x+y=y+x, for x,y∈ℜ(iii)
There exists0∈ℜ such that x+0=x, for any x∈ℜ(iv)
For anyx∈ℜ, there exists a unique element x′∈ℜ, such that 0∈x+x′ (x′ is called the opposite of x and it is denoted by −x)(v)
z∈x+y implies that y∈−x+z and x∈z−y; that is, ℜ,+ is reversibleDefinition 6.
(see [12]). ℜ,+,⋅ is called a Krasner hyperring if it satisfies the following conditions:(1)
ℜ,+ is a canonical hypergroup(2)
ℜ,⋅ is a semigroup having 0 as a bilaterally absorbing element, that is, x⋅0=0⋅x=0, for all x∈ℜ(3)
y+z⋅x=y⋅x+z⋅x and x⋅y+z=x⋅y+x⋅z, for all x,y,z∈ℜExample 1.
(for more details, see [39, 40]). Let H=0,1. Then, H,⊕,⋅ is a Krasner hyperring where ⋅ is the usual multiplication, and the hyperaddition ⊕ is defined by(3)x⊕y=maxx,y,ifx≠y,0,x,ifx=y.Lemma 1 (see [41]).
A nonempty subsetA of a Krasner hyperring ℜ is a left (resp. r⋅x∈Aresp right) hyperideal if and only if(i)
x−y⊆A, for x,y∈A(ii)
r⋅x∈Arespx⋅r∈A, for x∈A,r∈ℜDefinition 7.
(see [42]). Let N be a hyperideal of a hyperring ℜ. Then, N=r∈ℜ:rn∈N, for some n∈ℕ.
Letℜ be a hyperring. For a∈ℜ, we define anna=r∈ℜ:ra=0. If anna=0 (resp., anna≠0), a is said to be regular (resp., zero divisor). We use the notion rℜresp,zdℜ to denote the set of all regular elements (resp., zero divisors) [43]. If N is a hyperideal of ℜ and A a subset of ℜ, then we denote N:A=x∈ℜ,x⋅A⊆N. It is clear that 0:A=annA.Definition 8.
(see [37]). A proper hyperideal N of a commutative Krasner hyperring ℜ is called an r-hyperideal (pr-hyperideal), if a⋅b∈N and anna=0 implies that b∈N (bn∈N, for some n∈ℕ), for any a,b∈ℜ.
MinN is the set of all minimal prime hyperideals which contain N. Min0 is stated by Minℜ.Definition 9.
(cf. [44]). A hyperring ℜ satisfies the following:(i)
Property A: if any finitely generated hyperideal,N⊆zdℜ has nonzero annihilator(ii)
Annihilator condition: if for any finitely generated hyperidealN of ℜ, there exists an element a∈ℜ such that annN=anna(iii)
Strong annihilator condition (briefly s.a.c.): if for any finitely generated hyperidealN of ℜ, there exists an element a∈N such that annN=anna
N is a z0-hyperideal if a∈N, b∈ℜ and anna=annb imply that b∈N.
socℜ denotes the sum of all minimal hyperideals of ℜ. The socle of a reduced hyperring ℜ is called socℜ=⊕i∈Aeiℜ, where ei:i∈A is the set of idempotents of ℜ [45].
LetA be a subset of ℜ and a∈A such that a∈aℜa. Then, a is called von Neumann regular element. Therefore, if all of the elements are von Neumann regular, A or ℜ is called von Neumann regular [46].
A two-sided hyperidealN in a semihypergroup ℜ is called right pure hyperideal if, for each a∈N, there exists b∈N such that a∈ab [47]. Similarly, a hyperideal is called pure hyperideal in a Krasner hyperring, if for each a∈N, there exists b∈N such that a=ab.
A hyperringℜ is called a reduced hyperring if there are no nilpotent elements in ℜ. If xn=0 for x∈ℜ,n∈ℕ, then x=0 [48].
Letf:ℜ⟶S be a homomorphism, N be a hyperideal of ℜ, and M be a hyperideal of S. Then, the hyperideal fN is said to be the extension of N, and it is denoted by Ne. The hyperideal f−1M is called contraction of M and denoted by Mc [26]. The mapping φ:ℜ⟶S−1ℜ given by a=a/1 is a homomorphism. If N is a hyperideal of ℜ, then φN=S−1N=i/s∈S−1ℜ:i∈N,s∈S is also a hyperideal of S−1ℜ [10]. If S=rℜ, then the hyperring S−1ℜ is called quotient hyperring which is denoted by Qℜ.


## 3.r-Hyperideals

In this section, we examine some properties ofr-hyperideals in commutative hyperrings. We compare r-hyperideals with prime and maximal hyperideals. Throughout the section, ℜ,+,⋅ is a commutative Krasner hyperring.Initially, let us give some examples for better understanding.Example 2.
Clearly,ℤ6 is a Krasner hyperring with the usual addition and multiplication, and 0^,2^,4^=N is a proper hyperideal.ann5^=0 and 0^⋅5^=0^∈N implies that 0^∈Nann5^=0 and 2^⋅5^=4^∈N implies that 2^∈Nann5^=0 and 4^⋅5^=2^∈N implies that 4^∈N
ThenN is an r-hyperideal of ℤ6.Example 3.
(see [49]). Let H=0,1,a and B=0,a. Then, H,+,⋅ is a Krasner hyperring with the hyperaddition and multiplication defined by
Clearly,B is an r-hyperideal of H.Example 4.
LetR=0,a,b,c be a set with the hyperaddition ⊕ and multiplication ⊙ defined as follows:
Then,R,⊕,⊙ is a Krasner hyperring [48]. It can be easily seen that I1=0,b,I2=0,c,I3=0,b,c are r-hyperideals.
Our main results regardingr-hyperideals are the followings.Theorem 1.
Letℜ be a hyperring and N be a hyperideal of ℜ. Then, the following statements are equivalent:(a)
N is an r-hyperideal(b)
a⋅ℜ∩N=a⋅N, for any a∈rℜ(c)
N=N:a, for any a∈rℜ/N(d)
N=Mc, where M is a hyperideal of QℜProof.
a⇒b Let N be an r-hyperideal and a be a regular element. Suppose that x∈a⋅ℜ∩N. Then, x∈a⋅ℜ and x∈N. Thus, x=a⋅a′, for a′∈ℜ. Since x=a⋅a′∈N, anna=0 and N is an r-hyperideal, a′∈N. Hence, a⋅a′∈a⋅N and a⋅ℜ∩N⊆a⋅N.
Therefore, for everya∈ℜ, a⋅N⊆a⋅ℜ∩N, a⋅ℜ∩N=a⋅N.
b⇒c We know that N⊆N:a, for every a∈ℜ. Let a be regular, a∉N and x∈N:a. Hence, anna=0 and x⋅a∈N. From (b), a⋅x∈a⋅ℜ∩N and a⋅x∈a⋅N. This implies that a⋅x=a⋅y, for y∈N. Since anna=0, then x=y∈N. Hence, x∈N. Thus, N:a⊆N.
c⇒d Let S be the set of regular elements and φ:ℜ⟶Qℜ be a natural homomorphism. We know that N⊆φ−1M=Mc, for M a hyperideal of Qℜ. Suppose x∈φ−1M. Since φx=x/s∈S−1ℜ, then s⋅x∈N, for s∈S. From (c), x∈N:s=N.
d⇒a Let a⋅x∈N and anna=0. We have a⋅x/1=a/1⋅x/1∈M, and since a is regular, then there exists 1/a which is the inverse of a/1 in S. Thus, a/1⋅x/1⋅1/a∈M and x/1∈M. Hence, x∈N and so N is an r-hyperideal.Corollary 1.
The following statements hold:(a)
The zero hyperideal is anr-hyperideal(b)
The intersection ofr-hyperideals is an r-hyperideal(c)
WhenN is an r-hyperideal, N⊆zdℜ(d)
Everyr-hyperideal is a pr-hyperideal(e)
A prime hyperideal is anr-hyperideal if and only if it consists all of zerodivisors. As a result, every minimal prime hyperideal is an r-hyperideal(f)
LetN be an r-hyperideal, S⊆ℜ and S⊈N. Then, N:S is an r-hyperideal. Particularly, annx and annS are always r-hyperideals.(g)
Every minimal hyperideal of a reduced hyperring is anr-hyperideal(h)
Every pure hyperideal and every von Neumann regular hyperideal arer-hyperideals(i)
Suppose thatℜ satisfies the s.a.c. and N is a hyperideal of ℜ.N is an r-hyperideal if and only if for every hyperideal J and K of ℜ such that J is finitely generated, whenever J⋅K⊆N and annJ=0, then K⊆N(j)
The sum of twor-hyperideals may not be an r-hyperidealProof.
(b) Suppose thatI1,I2,…,In are r-hyperideals of ℜ. Let r⋅x∈∩i=1nIi and annr=0. Then, r⋅x∈Ii. Since every Ii is r-hyperideal, then we have x∈Ii. Hence, x∈∩i=1nIi., which means the intersection of r-hyperideals is also r-hyperideal.(c) LetN be an r-hyperideal of ℜ and N⊈zdℜ. Then, there exists a regular element r in N. Now, let e be identity element of ℜ. In this way, e⋅r∈N. Since N is an r-hyperideal, then e∈N. This is a contradiction to N’s being proper hyperideal.(d) If a prime hyperidealN is an r-hyperideal, then it consists all of zerodivisors from c. For the converse, suppose that N is prime and N⊆zdℜ. Let r⋅x∈N and annr=0. Since N is prime, then r∈N or x∈N. Since we assume N⊆zdℜ, then there is no regular element in N and so r∉N. Then, we get x∈N. We conclude that N is an r-hyperideal.(i) Assume thatN is an r-hyperideal, J is finitely generated hyperideal, and K is a hyperideal of ℜ such that J⋅K⊆N and annJ=0. Then, there exists a b∈J such that annJ=annb and so annb=0. For k∈K, we have b⋅k∈J⋅K⊆N. Since N is an r-hyperideal, then k∈N. That’s why K⊆N. Conversely, assume that for every hyperideal J and K of ℜ such that J is finitely generated, whenever J⋅K⊆N and annJ=0, then K⊆N. Let a⋅ℜ=J and b⋅ℜ=K. Then, J⋅K=a⋅b⋅ℜ⊆N and anna=0. Therefore, b⋅ℜ⊆N because of K⊆N. Thus, b∈N, and we get the desired result.(j) To prove that the sum of twor-hyperideals may not be an r-hyperideal, we give the following example:Example 5.
LetG,⊙ be a group and H=G∪0,u,v, where 0 is an absorbing element under multiplication and u,v are distinct orthogonal idempotents withu⊙v=v⊙u=0;u⊙u=uv⊙v=v;a⊙0=0⊙a=0; for all a∈Hu⊙g=g⊙u=u;v⊙g=g⊙v=v; for all g∈G
If we define the hyperaddition⊕ on H as follows:a⊕0=0⊕a=a;a⊕a=0,a, for all a∈Ha⊕b=b⊕a=H/0,a,b, for all a,b∈H/0 and a≠b
Then,H,⊕,⊙ is a Krasner hyperring [41]. Consider I=0,u and J=0,v. It can be easily seen that I,J are r-hyperideals, while I⊕J=H is not an r-hyperideal.
We omit the rest of proof since it is obvious.Remark 1.
LetK and L be hyperideals of Qℜ. We know that Kc⋅Lc⊆K⋅Lc and Kc+Lc⊆K+Lc. If N and M are r-hyperideals of ℜ, then by Theorem 1(d),N=Kc and M=Lc, for some hyperideals K and L in Qℜ. It follows that(a)
N⋅M is an r-hyperideal of ℜ if and only if K⋅Lc⊆Kc⋅Lc (essentially K⋅Lc=Kc⋅Lc)(b)
N+M is an r-hyperideal of ℜ if and only if K+Lc⊆Kc+Lc (essentially K+Lc=Kc+Lc)Lemma 2.
Letℜ be a hyperring and N be a hyperideal. Then, the following statements hold:(a)
N is an r-hyperideal if and only if for any J, K hyperideals of ℜ with J∩rℜ≠∅ and J⋅K⊆N, then K⊆N(b)
Assume thatN⊆zdℜ is not an r-hyperideal. There exist hyperideals J and K such that J∩rℜ≠∅, N⊊J, K, and J⋅K⊆NProof.
(a)
LetN be an r-hyperideal, J and K be hyperideals of ℜ such that J∩rℜ≠∅ and J⋅K⊆N. Let a∈J∩rℜ and b∈K. Since J and K are hyperideals, then we can take a⋅ℜ⊆J and b⋅ℜ⊆K. By our assumption, a⋅b⋅ℜ⊆N. Since J∩rℜ≠∅, then let we take anna=0. Then N is an r-hyperideal and so b⋅ℜ⊆N. Thus, K⊆N.Conversely, assume thatJ∩rℜ≠∅ and J⋅K⊆N,K⊆N. Let a⋅ℜ⊆J and b⋅ℜ⊆K. Then, a⋅b⋅ℜ⊆N. Let a∈J∩rℜ. So anna=0. At the same time, since K⊆N, then b⋅ℜ⊆N.(b)
Assume thatN⊆zdℜ is not an r-hyperideal. Then, there exist r∈rℜ and x∉N such that r⋅x∈N. We have annr=0. Let J=N:x and K=N:J. It follows r∈J. Since annr=0, then J∩rℜ≠∅ and r∉N. Since N⊆zdℜ, then annx≠0. Therefore, x∈K. Thus, there exist b,c∈ℜ such that b⋅x∈N and c⋅J⊆N. Hence, b⋅x⋅c⋅J⊆N. Then, x⋅J⋅K⊆N and J⋅K⊆N.Proposition 1.
(a)
Letℜ be a hyperring and N be a hyperideal of ℜ with N∩rℜ≠∅. If J and K are r-hyperideals of ℜ such that N⋅J=N⋅K or N∩J=N∩K, then J=K.(b)
Letℜ be a hyperring and N,M be hyperideals of ℜ with M∩rℜ≠∅. If N⋅M is r-hyperideal of ℜ, then N=N⋅M. In addition, N is an r-hyperideal.Proof.
(a)
LetJ and K be r-hyperideals of ℜ and N⋅J=N⋅K. From Lemma 1, since N⋅J=N⋅K⊆K, then J⊆K. Therefore, K⊆J. Then, J=K.(b)
LetN⋅M be r-hyperideal of ℜ and M∩rℜ≠∅. Since N⋅M⊆M⋅N, then from Lemma 1, N⊆N⋅M. Therefore, obviously, N⋅M⊆N. Then, N=N⋅M.Theorem 2.
LetI1,I2,…,In be prime hyperideals of ℜ, which are not comparable. If ∩i=1nIi is an r-hyperideal, then Ii is an r-hyperideal, for i=1,…,n.Proof.
Letr,x∈ℜ such that r⋅x∈Ii and annr=0. Let y∈∩j≠iIj/Ii. Then, r⋅x⋅y∈∩i=1nIi. We get that x⋅y∈∩i=1nIi, since ∩i=1nIi is an r-hyperideal and annr=0. Thus, x⋅y∈Ii. We conclude that x∈Ii, since y∉Ii and Ii’s are prime. Hence, Ii is an r-hyperideal.
Letρ:ℜ⟶S be a homomorphism. We investigate whether the image of an r-hyperideal and the inverse image of an r-hyperideal are r-hyperideal.Theorem 3.
Letρ:ℜ⟶S be a good epimorphism such that Kerρ⊆N. If N is an r-hyperideal of ℜ,+,⋅, then ρN is an r-hyperideal of the hyperring S,⊕,⊙.Proof.
Obviously,ρN is hyperideal of S. Let b1⊙b2∈ρN and annb1=0S, for b1,b2∈S. Since ρ is onto, then there exist a1,a2∈ℜ such that b1=ρa1 and b2=ρa2. Then, b1⊙b2=ρa1⊙ρa2=ρa1⋅a2=ρx∈fN, for some x∈N.
0∈ρa1⋅a2⊖ρx=ρa1⋅a2−x. Then, there exists t∈a1⋅a2−x such that ρt=0. We have a1⋅a2∈t+x⊆Kerρ+N⊆N+N⊆N. Thus, a1⋅a2∈N.
Letanna1≠0. Then, there exists 0≠c∈ℜ such that a1⋅c=0ℜ. So, ρa1⋅c=ρa1⊙ρc=ρ0ℜ=0S. Since ρc≠0S, then annρa1=annb1=0S, and this is a contradiction. This implies that anna1=0. Since N is an r-hyperideal, then a2∈N. Therefore, b2=ρa2∈ρN.Theorem 4.
Letρ:ℜ⟶S be a good monomorphism. If M is an r -hyperideal of S,⊕,⊙, then ρ−1M=Mc is an r-hyperideal of ℜ,+,⋅.Proof.
ρ−1M is a hyperideal [37]. Let r⋅x∈ρ−1M and annr=0. Then, ρr⊙ρx=ρr⋅x∈M. If annρr≠0, then there exists 0≠ρs∈M such that ρr⊙ρs=ρr⋅s=0S. This means that there exists a 0≠s∈ρ−1M such that r⋅s=0ℜ. Then, annr≠0. This is a contradiction, so annρr=0. Since M is an r-hyperideal, then ρx∈M and therefore x∈ρ−1M.Theorem 5.
Letℜ be a hyperring. The following statements are equivalent:(a)
ℜ is a hyperdomain(b)
The onlyr-hyperideal of ℜ is the zero hyperideal(c)
annx⋅y=annx∪anny, for each x,y∈ℜProof.
a⇒b Let us suppose that ℜ is hyperdomain and 0≠N is a proper hyperideal of ℜ. Then, there exists 0≠r∈N. Since ℜ is a hyperdomain, then we have annr=0. This contradicts the hyperideal of being proper. Then, the zero hyperideal is the only r-hyperideal.b⇒c From Corollary 1 (f), annx is an r-hyperideal. Assume that the zero hyperideal is the only r-hyperideal of ℜ. Hence, annx=0. So annx⋅y=annx∪anny=0, for every x,y∈ℜ.c⇒a Let x⋅y=0, x,y∈ℜ. Thus, annx⋅y=ann0=annx∪anny=ℜ. This means 1ℜ∈annx∪anny. Then, x=0 or y=0. Hence, ℜ is a hyperdomain.Proposition 2.
(a)
Letℜ be a hyperring and x,y∈ℜ with 1ℜ∈x+y. Then, N=annx+anny is an r-hyperideal.(b)
Letℜ be a reduced hyperring, I∈Minℜ and e∈ℜ be an idempotent element. We infer that N=I+anne is an r-hyperideal.Proof.
(a)
Leta⋅b∈N and anna=0. Then, there exists r∈annx and s∈anny such that a⋅b∈r+s. Obviously, r⋅x=0 and s⋅y=0. We get that a⋅b⋅x⋅y∈r+s⋅x⋅y=r⋅x⋅y+s⋅x⋅y=y⋅x⋅r+x⋅y⋅s=b⋅0+x⋅0=0 and by our assumption, b⋅x⋅y=0. Thus, b⋅x∈anny or b⋅y∈annx. Hence, b=b⋅1ℜ∈b⋅x+y=b⋅xb⋅y⊆annx+anny=N and therefore b∈N.(b)
Letr⋅x∈N and annr=0. We have r⋅x∈a+b such that a∈I and b∈anne. Since I∈Minℜ, then I is an r -hyperideal. There is y∉I such that a⋅y=0. e⋅y⋅r⋅x∈e⋅y⋅a+b=e⋅y⋅a+e⋅y⋅b=0. Since ℜ is a reduced hyperring, then every idempotent element is 0. By our assumption, e⋅y⋅x∈I. Since y∉I, then e⋅x∈I. We write x∈x+e⋅x−e⋅x, x∈e⋅x+x−e⋅x=e⋅x+x⋅1ℜ−e⊆I+anne=N. Then, N is an r-hyperideal.Corollary 2.
(a)
If there is a hyperidealK of ℜ such that annN+annM=annK, for N,M hyperideals, then annN+annM is r-hyperideal.(b)
The direct sum ofr-hyperideals is r-hyperideal. Therefore, if N=M⊕K and N is r-hyperideal, then M and K are r-hyperideals(c)
Letℜ be a reduced hyperring. Then, socℜ is an r-hyperideal. Therefore, there is a hyperideal M of ℜ such that socℜ=Mc.Proposition 3.
LetN be a hyperideal of ℜ. N is a pr-hyperideal if and only if N is r-hyperideal.Proof.
Similarly with [33], let N be pr-hyperideal, x⋅y∈N and annx=0. There is n∈ℕ such that x⋅yn∈N. Hence, xn⋅yn∈N and annxn=0. Since N is pr-hyperideal, then ynm∈N, for m∈ℕ=0. Obviously, 0 is r-hyperideal. Then, nilℜ is also r-hyperideal.
From Theorem1 and Proposition 3, we obtain the next corollary.Corollary 3.
LetN be a hyperideal of ℜ. The following statements are equivalent:(a)
N is pr-hyperideal(b)
r⋅ℜ∩N=r⋅N, for any r∈rℜ(c)
N=N:r, for any r∈rℜ and r∉N(d)
N=Mc, M is a primary hyperideal of QℜTheorem 6.
Everyz0-hyperideal is an r-hyperideal.Proof.
Similar to [33], let x⋅y∈N and annx=0. Then, annx⋅y=anny. Since N is z0-hyperideal, then y∈N and N are an r-hyperideal.Theorem 7.
Every hyperideal which contains zero divisors is contained by a primer-hyperideal.Proof.
It is trivial.Theorem 8.
IfN is a hyperideal and I∈MinN in hyperring ℜ, then I is an r-hyperideal.Proof.
Letx⋅y∈I and annx=0. There exists a∉I and n∈ℕ such that a⋅x⋅yn=a⋅xn⋅yn∈N. Since N is r-hyperideal and annxn=0, then a⋅yn∈N⊆I. Since a∉I, then yn∈I. Thus, y∈I.Proposition 4.
LetN be an r-hyperideal and N⊆M. If M/N is r-hyperideal of ℜ/N, then M is r-hyperideal of ℜ.Proof.
Let ber-hyperideal of ℜ/N, a+N⋅b+N⊆M/N and anna+N=0 for a+N,b+N∈ℜ/N.
a+N⋅b+N⊆a⋅b+N+N⋅b+N⊆a⋅b+a⋅N+N⋅b+N⊆a⋅b+N⊆M+N. So, a⋅b∈M. Since anna=0 and b∈M, then M is an r-hyperideal.
Remember that ifM and ℜ are the only hyperideals of ℜ that contain M, a proper hyperideal M of ℜ is said to be a maximal hyperideal of ℜ. Moreover, in a commutative unitary hyperring, each maximal hyperideal is a prime hyperideal (for more details, see [41]). In the following, we investigate the maximal hyperideal and r-hyperideal relation.Proposition 5.
Every maximalr-hyperideal is a prime hyperideal.Proof.
Obvious from [33].Proposition 6.
If every prime hyperideal ofℜ is an r-hyperideal, every maximal hyperideal of ℜ is an r-hyperideal.Proof.
Suppose thatM is a maximal hyperideal of ℜ. We know that every maximal hyperideal is a prime hyperideal. Let a⋅b∈M with anna=0. Then, b∈M:a. Since M is maximal, then M=M:a and b∈M.Proposition 7.
Letℜ be a reduced hyperring satisfying the Property A. Then, every maximal r-hyperideal is an z0-hyperideal.Proof.
LetI be a maximal r-hyperideal, I⊆zdℜ. There exists a z0-hyperideal J such that I⊆J. Then, J is an r-hyperideal. Since I is maximal, then I=J. Thus, I is an z0-hyperideal.


## 4. Generalizations ofr-Hyperideals

In this section, we introduce a generalization ofr-hyperideals to commutative Krasner hyperrings. Idℜ denotes the hyperideals of ℜ. Initially, we give the definition of ϕ− prime and ϕ− primary hyperideals. Similarly, we give the definitions of ϕ−r-hyperideal, ϕ−pr-hyperideal, ϕ− pure hyperideal, and ϕ− von Neumann regular hyperideal in ℜ. Afterwards, we investigate some properties of ϕ−r-hyperideal of ℜ.Definition 10.
Letℜ be a Krasner hyperring, ϕ:Idℜ⟶Idℜ∪∅ be a function and ∅≠N∈Idℜ. Then, N is said to be a ϕ− prime (resp. ϕ− primary) hyperideal of ℜ if whenever r,s∈ℜ and r⋅s∈N−ϕN, then r∈N or s∈N (resp. r∈N or sn∈N).Definition 11.
Letℜ be a Krasner hyperring, N be a proper hyperideal of ℜ and ϕ:Idℜ⟶Idℜ∪∅ be a function. N is called ϕ−r-hyperideal (resp., ϕ−pr-hyperideal) if a⋅b∈N−ϕN with anna=0 implies that b∈N (resp., bn∈N), for a,b∈ℜ.Example 6.
LetJ be a ϕ−r-hyperideal of ℜ. Then,(i)
IfϕJ=∅, for all J∈Idℜ, then J is called ϕ∅−r-hyperideal of ℜ (r-hyperideal).(ii)
IfϕJ=0, for all J∈Idℜ, then J is called ϕ0−r-hyperideal of ℜ (weakly r-hyperideal).(iii)
IfϕJ=J, for all J∈Idℜ, then J is called ϕ1−r-hyperideal of ℜ (any hyperideal).(iv)
IfϕJ=Jn, for all J∈Idℜ and n≥2, then N is called ϕn−r-hyperideal (n-almost r-hyperideal). If n=2, then J is called almost r-hyperideal of ℜ.(v)
IfϕJ=∩i=1∞Ji, for all J∈Idℜ, then J is called ϕω−r-hyperideal (ω−r-hyperideal) of ℜ.Definition 12.
Letℜ be a Krasner hyperring, N be a proper hyperideal of ℜ and ϕ:Idℜ⟶Idℜ∪0 be a function.(1)
N is called ϕ− pure hyperideal if there exists b∈N such that a=a⋅b, for every a∈N−ϕN(2)
N is called ϕ− von Neumann regular hyperideal if there exists b∈N such that a=a2⋅b, for every a∈N−ϕNTheorem 9.
LetN∈Idℜ. Then, the following statements hold:(i)
Letψ1 and ψ2 be two functions. ψ1,2:Idℜ⟶Idℜ∪∅ such that ψ1≤ψ2. If N is a ψ1−r-hyperideal, then N is a ψ2−r-hyperideal(ii)
Everyϕ−r-hyperideal is a ϕ−pr-hyperideal(iii)
N is an r-hyperideal ⇒N is a weakly r-hyperideal ⇒N is an ω−r-hyperideal ⇒N is an n+1-almost r-hyperideal ⇒N is an n-almost r-hyperideal ⇒N is an almost r-hyperideal(iv)
Everyϕ− pure hyperideal is a ϕ−r-hyperideal(v)
Everyϕ− von Neumann regular hyperideal is a ϕ−r-hyperidealProof.
Straightforward.Theorem 10.
Letℜ be a Krasner hyperring, N be a proper hyperideal of ℜ. Then, the following statements are equivalent:(i)
N is ϕ−r-hyperideal(ii)
N:r=N∪ϕN:r, r∈rℜ\N(iii)
N:r=N or N:r=ϕN:r, r∈rℜ/NProof.
i⇒ii We know that N⊆N:r and ϕN:r⊆N:r. Let s∈N:r. Then, r⋅s∈N. If r⋅s∈ϕN, then s∈ϕN:r. On the other hand, if r⋅s∉r⋅s∉fa, since N is ϕ−r-hyperideal, then s∈N.ii⇒iii It is evident.iii⇒i Let r,s∈ℜ such that r⋅s∈N−ϕN and r be regular. Then, N:r=N or N:r=ϕN:r.Case 1.
.Assume thatN:r=N. Since r⋅s∈N, then s∈N:r=N.Case 2.
.Assume thatN:r=ϕN:r. Since r⋅s∈N, then s∈N:r. Thus, r⋅s∈ϕN. This is a contradiction.Definition 13.
LetN be a proper hyperideal of ℜ, J and K be hyperideals of ℜ such that J⋅K⊆N, J⋅K⊈ϕN and annJ=0 implies that K⊆N. Then, N is called a strongly ϕ−r-hyperideal.Theorem 11.
Every stronglyϕ−r-hyperideal is an ϕ−r-hyperideal.Proof.
.Similar to [36], let a,b∈ℜ such that a⋅b∈N−ϕN and anna=0. Then, a⋅b⊆N−ϕN. Obviously, anna=0. Since N is strongly ϕ−r-hyperideal, then <b>⊆N. Therefore, b∈N.Theorem 12.
Letℜ satisfy the s.a.c., N be a proper hyperideal of ℜ and ϕN be an r-hyperideal of ℜ. N is ϕ−r-hyperideal if and only if for every hyperideal J and K of ℜ such that J is finitely generated, whenever J⋅K⊆N, J⋅K⊈ϕN, and annJ=0, implies that K⊆N.Proof.
⇒: Let N be a ϕ−r-hyperideal, ϕN be an r-hyperideal, J be finitely generated, J and K be hyperideals of ℜ such that J⋅K⊆N and J⋅K⊈ϕN. Assume that K⊈N. Then, there exists an element 0≠b∈K−N. Since ℜ satisfy the s.a.c and J is finitely generated, then there is an element a∈J with annJ=anna=0. Let a⋅b∈N. If a⋅b∉ϕN, since N is ϕ−r-hyperideal, then b∈N which is a contradiction. If a⋅b∈ϕN, since ϕN is an r-hyperideal, then b∈ϕN which is a contradiction. Thus, K⊆N.
⇐: It is obvious.
Now, we investigate the relation between the hyperidealN and the set of all zero divisors, while ϕN is an r-hyperideal of ℜ.Theorem 13.
LetϕN be an r-hyperideal of ℜ and N be a proper hyperideal of ℜ. If N is ϕ−r-hyperideal, then N⊆zdℜ.Proof.
LetN⊈zdℜ. Then, there exists an element a∈N−zdℜ. So, anna=0. We have a=a⋅1ℜ∈N. If a∉ϕN, then 1ℜ∈N, which is a contradiction. If a∈ϕN, then 1ℜ∈ϕN⊆N, since ϕN is an r-hyperideal.Theorem 14.
LetϕN be an r-hyperideal of ℜ and N be a prime hyperideal of ℜ. N is ϕ−r-hyperideal if and only if N⊆zdℜ.Proof.
LetϕN be an r-hyperideal of ℜ and N be a prime hyperideal.
⇒: It is obvious from the previous theorem.
⇐: Let a⋅b∈N−ϕN and anna=0. Since N is a prime hyperideal, a∈N or b∈N. Since N⊆zdℜ, then a∉N. Therefore, b∈N. □
In a hyperidealN, nilpotent elements are also zero divisors. So we can change the theorem as if N is a hyperideal of which elements are nilpotent, then N is ϕ−r-hyperideal if and only if N is ϕ-primary hyperideal.Theorem 15.
IfN is ϕ−r-hyperideal, then N−ϕN⊆zdℜ.Proof.
Let us suppose thatN−ϕN⊈zdℜ. Then, there exists an element a in N−ϕN such that anna=0. Since a=a⋅1∈N−ϕN and N is ϕ−r-hyperideal, then 1∈N. That is a contradiction.Theorem 16.
LetϕN be an r-hyperideal. N is ϕ−r-hyperideal if and only if N is an r-hyperideal.Proof.
⇒: Let a⋅b∈N and anna=0. If a⋅b∈ϕN, since ϕN is an r-hyperideal, then b∈ϕN⊆N. If a⋅b∉ϕN, since N is ϕ− r-hyperideal, then b∈N.
⇐: It is obvious.
Similar to [36], if N is a hyperideal of ℜ, we define ϕN:Idℜ/N⟶Idℜ/N∪∅ by ϕNM/N=ϕM+N/N for each hyperideal N⊆M. If ϕN=∅, ϕNM/N=∅. We see that ϕNM/N⊆M/N.Theorem 17.
Suppose thatN⊆rℜ and M are hyperideals in ℜ such that N⊆M. If M is ϕ−r-hyperideal of ℜ, then M/N is ϕN−r-hyperideal of ℜ/N.Proof.
Suppose thatr+N⋅s+N⊆M/N−ϕMM/N=M/N−ϕM+N/N and annℜ/Nr+N=0. Then, r⋅s∈M−ϕM and as N⊆rℜ, annℜr=0. Since M is ϕ−r-hyperideal, then s∈M. Therefore, s+N⊆M/N.Theorem 18.
Suppose thatN is an r-hyperideal and M is a hyperideal in ℜ such that N⊆M. If M/N is r-hyperideal of ℜ/N, then M is ϕ−r-hyperideal of ℜ.Proof.
Suppose thatr⋅s∈M−ϕM with annr=0. If r⋅s∈N, since N is an r-hyperideal, then s∈N⊆M. If r⋅s∈M−N, then r⋅s+N=r+N⋅s+N⊆M/N. In addition, since annr=0, then annr+N=0. Therefore, since M/N is r-hyperideal of ℜ/N, then s+N⊆M/N. Hence, s∈M, as desired.
Letϕ:Idℜ⟶Idℜ∪0 be a function. If N and M are hyperideals of ℜ such that N⊆M implies that ϕN⊆ϕM, then ϕ is said to preserve the order.Theorem 19.
LetA be a subset of ℜ, A≠∅ and ϕ preserves the order. If N is ϕ−r-hyperideal such that A⊈N, then N:A is ϕ−r-hyperideal.Proof.
Suppose thata⋅b∈N:A−ϕN:A and anna=0. Since ϕ preserves order, then ϕN⊆ϕN:A. We have a⋅b⋅A⊆N−ϕN. Therefore, b⋅A⊆N and b∈N:A.Corollary 4.
LetNii∈Λ directed collection of ascending chain ϕ−r-hyperideals of ℜ and ϕ preserves the order. Then, N=∪N∈ΛNi is ϕ−r-hyperideal.Proof.
Leta⋅b∈N−ϕN and anna=0. Then, a⋅b∈Nk and a⋅b∉ϕNN, for k∈Λ. Since Nk is ϕ−r-hyperideal, then b∈Nk⊆N.


## 5. Conclusion

In this paper, some properties ofr-hyperideals are investigated in the commutative Krasner hyperrings. For r-hyperideals and pr-hyperideals, many characterizations are introduced, and some examples are given. The theoretical point of view of ϕ−r-hyperideals which is the generalization of ϕ−r-ideal is examined. After this study, one can introduce and examine them in Krasner m,n-hyperrings and do some studies on their properties. This approach can also be used to algebraic structures like fuzzy sets and Zariski topology. We suggest open problems with δ-r-hyperideals, 2-absorbing r-hyperideals, and r-subhypermodules.



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