# ϕ -δ-Primary Hyperideals in Krasner Hyperrings

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

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## Abstract

In this paper, we study commutative Krasner hyperrings with nonzero identity.ϕ-prime, ϕ-primary and ϕ-δ-primary hyperideals are introduced. The concept of δ-primary hyperideals is extended to ϕ-δ-primary hyperideals. Some characterizations of hyperideals are provided to classify them. The relation between ϕ-δ-primary hyperideals and other hyperideals is discussed.


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## Body

## 1. Introduction

In commutative ring theory, prime and primary ideals have a significant place. The importance of prime ideals encourages researchers to expand these concepts and find applications. Many different types of generalizations have been investigated by several authors, some of them [1–5]. Prime and primary ideals are generalized to ϕ -prime and ϕ-primary ideals. Let ℜ,+,. be a commutative ring with nonzero identity. Denote the set of all ideals of ℜ by Lℜ (proper ideals of ℜ by L∗ℜ. Let ϕ be a function such that ϕ:Lℜ⟶Lℜ∪∅. Let N be a proper ideal of ℜ. N is called a ϕ-prime ideal [1], if ab∈N−ϕN, then either a∈N or b∈N for some a,b∈ℜ. By the way, N is called a ϕ-primary ideal when, if ab∈N−ϕN, then a∈N or bk∈N for some a,b∈ℜ,k∈ℕ [3, 4]. The image of ideal, ϕN, can be equal 0, ∅,N,N2, Nn, Nw (w denotes the intersection of ideals of Ni). A proper ideal N of ℜ is called weakly prime (primary) ideal respectively in [6] ([7]) if 0≠ab∈N, for some a,b∈ℜ, then a∈N or b∈N (bk∈N for some k∈ℕ). Anderson generalized it in [1], where N is weakly ϕ-prime ideal when ϕN=0. Zhao [8] introduced δ-primary ideal as an expansion of an ideal, δ:Lℜ⟶Lℜ is a function that meets the following requirements: iN⊆δN, for all ideals N of ℜ,ii If N⊆M, where N and M are ideals of ℜ, then δN⊆δM,iiiδK∩L=δK∩δL for all ideals K,L of ℜ. Entire of the δ ideal expansions provides the property δ2=δ, which is δδN=δN for all ideal N of ℜ [8]. A. Jaber chose ϕ such a reduction function in [9], which satisfies the following requirements: iϕN⊆N, for all ideals N of ℜ, ii If N⊆M, where N and M are ideals of ℜ, then ϕN⊆ϕM. He obtained generalization of ϕ-δ-primary ideal by combining these two concepts. Let N be an ideal of ℜ, δ be an ideal expansion and ϕ be an ideal reduction [9]. N is called ϕ- δ-primary if ab∈N−ϕN, then either a∈N or b∈δN, for all a,b∈ℜ. Some results on φ−δ-primary ideals can be found in [10, 11].The theory of hyperstructures was innovated by Marty in 1934 [12]. He defined hypergroupoid G,° for G≠∅, P∗G represents family of nonempty subsets of G and °:G×G⟶P∗G is a binary hyperoperation. Let G,° be a hypergroupoid. G is a semihypergroup, if ∀a,b,c∈G, a°b°c=a°b°c, which means ∪u∈a°bu°c=∪v∈b°ca°v. If ∀a∈G, there exists e∈G such that a∈e°a∩a°e in another phrase a⊆e°a∩a°e, then e is called identity element. An identity element e is called a scalar identity if a=e°a∩a°e, for all a∈G. Let G,° be a semihypergroup. For ∀a∈G, if a°G=G°a=G, then G,° is called hypergroup. Let G,° be a hypergroup and ∅≠K be a subset of G. If a°K=K°a=K, for ∀a∈K, then K,° is called subhypergroup of G,° where ° is a binary hyperoperation on G. Let G,° be a hypergroup. If a°b=b°a, for ∀a,b∈G, then G,° is commutative hypergroup [12]. Mittas pioneered the theory of canonical hypergroups in [13–15]: Let ℜ≠∅. ℜ,+ is called a canonical hypergroup (+ is a hyperoperation) if the following axioms are satisfied: ia+b+c=a+b+c, for a,b,c∈ℜ;iia+b=b+a, for a,b∈ℜ; iii∃0∈ℜ such that a+0=a, for any a∈ℜ; iv for any a∈ℜ, there exists a unique element a′∈ℜ, such that 0∈a+a′ (a′ is called as the opposite of a and it is denoted by −a); vc∈a+b implies that b∈−a+c and a∈c−b, which means ℜ,+ is reversible. Hyperrings and hyperfields were introduced by Krasner in [16] using the canonical hypergroups. ℜ,+,⋅ is called Krasner hyperring if the following statements hold: iℜ,+ is a canonical hypergroup; iiℜ,⋅ is a semigroup having 0 as a⋅0=0⋅a=0, for all a∈ℜ; iiib+c⋅a=b⋅a+c⋅a and a⋅b+c=a⋅b+a⋅c, for all a,b,c∈ℜ. Also, Ameri and Norouzi [17] studied general commutative hyperrings. Corsini and Leoreanu [18] presented some of the most important results on hyperrings and illustrated some of the most recent and interesting applications, that is, those to geometry, graphs and hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras and C-algebras, artificial intelligence and probability. Dasgupta [19] investigated extensively the prime and primary hyperideals of multiplicative hyperrings with absorbing zero. Davvaz and Leoreanu-Fotea, in their monograph [20], presented the main results obtained in hyperring theory till the publication of it, but also an outline of applications of hyperstructures. Recently, Davvaz [21] published the first monograph on semihypergroup theory which covers most of the mathematical ideas and techniques required in the study of semihypergroups. Recently, in [22], r-hyperideals were mentioned as a generalization of r-ideals in commutative rings, and in [23], properties of r-hyperideals and some generalizations of them are investigated in the commutative Krasner hyperrings. Yeşilot et al. defined a hyperideal expansion and δ-primary hyperideal in [24]. Let ℜ,⊕,° be a commutative Krasner hyperring, with nonzero unit, δ:Lℜ⟶Lℜ function is defined as a hyperideal expansion function that meets the following requirements where Lℜ denotes all hyperideals of ℜ: iN⊆δN, for all N∈Lℜ, ii If N⊆M, where N, M∈Lℜ, then δN⊆δM. Given an expansion δ of a hyperideal N of ℜ is called δ-primary if a°b∈N, then a∈N or b∈δN, for all a,b∈ℜ. Ulucak [25] obtained some results on δ-primary and 2-absorbing δ-primary hyperideals.In this paper, our goal is to extend the concept ofδ-primary hyperideals to ϕ-δ-primary hyperideals in Krasner hyperrings and we intend to give some generalizations of hyperideals. Throughout this paper, ℜ,⊕,° will be a commutative Krasner hyperring with nonzero identity. We denote the set of all hyperideals of ℜ by Lℜ and the set of all proper hyperideals of ℜ by L∗ℜ. We take ϕ as a function ϕ:Lℜ⟶Lℜ∪∅ at the sections of generalizations of prime and primary hyperideals. Firstly, we define ϕ-prime and ϕ-primary hyperideals in Krasner hyperrings and we give some characterizations for prime and primary hyperideals. Let ϕ be a function such that ϕ:Lℜ⟶Lℜ∪∅ and N be a hyperideal of ℜ. N is said to be ϕ-prime (ϕ-primary) hyperideal of ℜ if a°b∈N−ϕN, then a∈N or b∈N (resp., bn∈N, for some n∈ℕ), for a,b∈ℜ. Among other things, we give some characterizations in Theorem 2 and Theorem 5 as main theorems. Then we define ϕ-δ-primary hyperideals in Krasner hyperrings and also give several characterizations (See; Theorem 7 and Theorem 11). In this case, difference is that, ϕ:Lℜ⟶Lℜ∪∅ is a reduction function if ϕN⊆N and N⊆M implies ϕN⊆ϕM for each N,M∈Lℜ. We investigate the attitude of ϕ-δ-primary hyperideal under homomorphism, in quotient ring, in Cartesian product and other cases (See; Theorem 10, Proposition 10, Proposition 9, Theorem 12).


## 2. Generalizations of Prime Hyperideals in Krasner Hyperrings

Throughout this section,ℜ,⊕,° is a commutative Krasner hyperring with nonzero identity. We denote the set of all hyperideals of ℜ by Lℜ. Initially, we give the definition of ϕ-prime hyperideal and some examples.Definition 1.
Letϕbe a function such thatϕ:Lℜ⟶Lℜ∪∅andNbe a hyperideal ofℜ.Nis said to be aϕ-prime hyperideal ofℜifa°b∈N−ϕN, thena∈Norb∈N, fora,b∈ℜ.Example 1.
Letℜbe a commutative Krasner hyperring. Consider the following functionsϕ:Lℜ⟶Lℜ∪∅defined as follows:
For anyN∈Lℜ,(i)
ϕ⊘N=∅.(ii)
ϕ0N=0.(iii)
ϕ2N=N2.(iv)
ϕnN=Nn, (for anyn≥2(v)
ϕwN=∩n=1∞Nn.(vi)
ϕ1N=N.
It is obvious thatϕ⊘≤ϕ0≤ϕw≤⋯≤ϕn+1≤ϕn≤ϕn−1≤⋯≤ϕ2≤ϕ1.Definition 2.
Letℜbe a hyperring andNbe a proper hyperideal ofℜ.(i)
Nis prime hyperideal if and only if it isϕ∅-prime hyperideal.(ii)
Nis weakly prime hyperideal if and only if it isϕ0-prime hyperideal.(iii)
Nis almost prime hyperideal if and only if it isϕ2-prime hyperideal.(iv)
Nisn-almost prime hyperideal if and only if it isϕn-prime hyperideal.(v)
Nisw-prime hyperideal if and only if it isϕw-prime hyperideal.Proposition 1.
LetNbe a proper hyperideal ofℜ.(1)
Ifσ1, σ2 are two functions with σ1≤σ2 such that σ1,σ2:Lℜ⟶Lℜ∪∅ and N is σ1-prime, then N is σ2-prime.(2)
iN is prime hyperideal ⇒N is weakly prime hyperideal ⇒N is w-prime hyperideal ⇒N is n+1-almost prime hyperideal ⇒N is n-almost prime hyperideal for any n≥2⇒N is almost prime hyperideal.iiN is w-prime hyperideal if and only if N is n-almost prime for any n≥2.Proof.
(1)
Let us assumeN is a σ1-prime hyperideal of ℜ. Take a°b∈N and a°b∉σ2N, for a,b∈ℜ. It follows a°b∉σ1N because of σ1≤σ2. Therefore a°b∈N−σ1N and on the assumption that N is σ1-prime hyperideal, then a∈N or b∈N. That means N is σ2-prime hyperideal of ℜ.(2)
i We can obtain it by the ordering of the ϕi′s in Example 1 with the proof of 1.ii Assume that N is w-prime. Let a°b∈N−ϕwN=N−∩n=1∞Nn. Then a°b∈N−ϕnN, since ϕw≤ϕn, from (1) N is ϕn-prime. It means N is n-almost prime, for all n≥2. Conversely, suppose that N is n -almost prime, for a°b∈N−Nn for all n≥2. Thus a°b∈N−∩n=2∞Nn. Hence a°b∈N−∩n=1∞Nn. Therefore a∈N or b∈N, since N is n-almost prime.Theorem 1.
Letϕ:Lℜ⟶Lℜ∪∅be a function andTbe a proper hyperideal ofℜsuch thatTis aϕ-prime hyperideal ofℜ. IfTis not prime, thenT2⊆ϕT. Hence at the same time it means ifT2⊈ϕT, thenTis prime.Proof.
Assume thatT2⊈ϕT. We need to prove that T is prime. Take x°y∈T for x,y∈ℜ. If x°y∉ϕT, then x°y∈T−ϕT. Since T is ϕ-prime, then x∈T or y∈T. Let us suppose x°y∈ϕT. We can presume x°T⊈ϕK, so x°m∉ϕT for some m∈T. Then x°y⊕m⊆T−ϕT. So x∈T or y⊕m⊆T, since T is ϕ-prime hyperideal and hence x∈T or y∈T. Now on we can assume that x°T⊆ϕT. (Similarly, y°T⊆ϕT) We can find some n,k∈T with n°k∉ϕT because of T2⊈ϕT. Then x⊕n°y⊕k⊆T−ϕT. Since T is ϕ-prime hyperideal, then x⊕n⊆T or y⊕k⊆T. So we have x∈T or y∈T. Hence T is prime hyperideal of ℜ.Corollary 1.
IfTis aϕ-prime hyperideal ofℜwithϕ≤ϕ3, thenTisw-prime hyperideal.Proof.
We know thatT is ϕ-prime hyperideal while T is prime, for every ϕ. Therefore T is w-prime hyperideal of ℜ. Assume that T is not prime. By Theorem 1, T2⊆T⊆T3. So ϕT=Tn for every n≥2 . Therefore T is n-almost prime hyperideal for every n≥2. Hence T is w-prime hyperideal of ℜ.
Some characterizations ofϕ-prime hyperideals are provided.Theorem 2.
LetNbe a proper hyperideal of the commutative Krasner hyperringℜand letϕ:Lℜ⟶Lℜ∪∅be a function. Then the following statements hold:(i)
Nisϕ-prime hyperideal ofℜ.(ii)
Fora∈ℜ−N,N:a=N∪ϕN:a.(iii)
Fora∈ℜ−N, N:a=NorN:a=ϕN:a.(iv)
For each hyperidealsKandLofℜsuch thatK°L⊆NandK°L⊈ϕN, we haveK⊆NorL⊆N.Proof.
i⇒ii Take a∈ℜ−N. Suppose that b∈N:a, then a°b∈N. If a°b∉ϕN, then b∈N since N is ϕ-prime hyperideal of ℜ. If a°b∈ϕN, then b∈ϕN:a. Hence N:a⊆N∪ϕN:a. Other side holds since the assumption of ϕN⊆N.ii⇒iii It is obvious because of N:a is a hyperideal of ℜ.iii⇒iv Let K and L be hyperideals of ℜ such that K°L⊆N−ϕN. Assume that K⊈N. Then there exists an element a∈K−N, by iii we have N:a=N or N:a=ϕN:a. If a°L⊈ϕN, then L⊈ϕN:a. Since iii holds, then L⊆N:a=N. We are done. Assume that a°L⊆ϕN. Since K°L⊈ϕN, then we can choose an element x∈K such that x°L⊆N−ϕN. If x∉N, then by iii, N:x=N or N:x=ϕN:x. Since L⊆N:x, but L⊈ϕN:x, then we conclude that L⊆N:x=N. Assume that x∈N. We have a⊕x⊆K−N, and also note that a⊕x°L⊆N−ϕN, since a°L⊆ϕN and x°L⊈ϕN. Then L⊆N:a⊕x=N which completes the proof.iv⇒i Let a°b∈N−ϕN. Then a°b⊆N, but a°b⊈ϕN. So a⊆N or b⊆N. Then a∈N or b∈N.
In Theorem1 we show that, if T is a ϕ-prime hyperideal and T is not prime, then T2⊆ϕT. In the following, a corollary of Theorem 2 is given, the proof of which is similar to Corollary 14 [1].Corollary 2.
LetTbe aϕ-prime hyperideal that is not prime. ThenT°ϕT⊆ϕT.Proof.
Leta∈ϕT. If a∈T, then a°T⊆T2⊆ϕT from Theorem 1. Let us suppose that a∉T. From the main Theorem 2, T:a=T or T:a=ϕT:a as T⊆T:a gives a°T⊆ϕT. Let us suppose T:a=T. Assume that an∈ϕT, but an−1∉ϕT. Then an∈T, so an−1∈T:a=T. Hence an−1∈T−ϕT, so a∈T, which is a contradiction.
In the following, we give a proposition about quotient and localization of a hyperring. LetS be a multiplicatively closed subset of a Krasner hyperring ℜ. Define ϕ:Lℜ⟶Lℜ∪∅ and ϕS:LℜS⟶LℜS∪∅ with ϕSM=ϕM∩ℜS. Also ϕSM=∅, where. ϕM∩ℜ=∅.
LetN,M be hyperideals of ℜ and N⊆M. Define ϕM:LR/M⟶LR/M∪∅ with ϕMN/M=ϕN⊕M/M. Also ϕMN/M=∅, where ϕN=∅.Proposition 2.
Suppose thatϕ:Lℜ⟶Lℜ∪∅is a function andTis aϕ-prime hyperideal ofℜ.(i)
IfMis a hyperideal ofℜwithM⊆T, thenT/MisϕM-prime hyperideal ofℜ/M.(ii)
LetSbe a multiplicatively closed subset ofℜwithT∩S=∅andϕTS⊆ϕSTS. ThenTSis aϕS-prime hyperideal ofℜS.Proof.
(i)
Letx,y∈ℜ. Suppose that x⊕M°y⊕M⊆T/M−ϕMT/M. So x°y⊕M⊆T/M−ϕT⊕M/M∈. Then x°y∈T−ϕT⊕M We find that x°y∈T−ϕT and then x∈T or y∈T. Therefore x⊕M⊆T/M or y⊕M⊆T/M. So T/M is ϕM-prime hyperideal of ℜ/M.(ii)
Leta/s°b/t∈TS−ϕSTS, for some a,b∈ℜ;s,t∈S. Then we have p°a°b∈T for some p∈S but q°a°b∉ϕT∩ℜ for every q∈S. Now if q°a°b∈ϕT, then a/s°b/t∈ϕTS⊆ϕSTS which is a contradiction. So p°a°b∈T−ϕT and since T is ϕ-prime hyperideal of ℜ, then we get either p°a∈T or b∈T. Hence a/s∈TS or b/t∈TS, since T∩S=∅. Thus TS is ϕS-prime hyperideal of ℜS.Theorem 3.
(i)
LetXandYbe commutative Krasner hyperrings andNbe a weakly prime hyperideal ofX. ThenM=N⊗Yθis aϕ-prime hyperideal ofℜ=X⊗Y, for allϕwithϕw≤ϕ≤ϕ1.(ii)
Supposeℜis a commutative Krasner hyperring andMis a finitely generated proper hyperideal ofℜ. AssumingMis aϕ-prime hyperideal withϕ≤ϕ3.ThenMis either weakly prime orM2≠0is idempotent andℜdecomposes asX⊗YwhereY=M2andM=N⊗YwithNbe weakly prime. As a resultMisϕ-prime for eachϕwithϕw≤ϕ≤ϕ1.Proof.
(i)
LetN be a weakly prime hyperideal of X. Then M=N⊗Y not necessarily to be a weakly prime hyperideal of ℜ=X⊗Y; actually M is weakly prime hyperideal if and only if M is prime hyperideal. Nevertheless M is a ϕ-prime hyperideal for each ϕ with ϕw≤ϕ. If N is prime, then M is prime and hence ϕ-prime for all ϕ. Assume that N is not prime. Then N2=0. Therefore M2=0⊗Y and hereby ϕwM=0⊗Y. Hence we can write M−ϕwM in this form M−ϕwM=N⊗Y−0⊗Y=N−0⊗Y. Let we take a1,a2°b1,b2=a1°b1,a2°b2∈M−ϕwM . It means a1°b1∈N−0, so a1∈N or b1∈N. Then a1,a2∈M or b1,b2∈M. Hence M is ϕw-prime, therefore ϕ-prime hyperideal.(ii)
IfM is prime hyperideal, then M is weakly prime hyperideal of ℜ . Suppose that M is not prime. From Theorem 1, M2⊆ϕM; and hence M2⊆ϕM⊆ϕ3M=M3. So M2=M3, it means M2 is idempotent. Since M2 is finitely generated, then M2=m for some idempotent m∈ℜ. Assume that M2=0. Then ϕM⊆M3=0. Hereby ϕM=0. Consequently M is weakly prime hyperideal of ℜ. Now suppose M2≠0. Take that Y=M2=ℜ°m and X=ℜ°1⊖m, so ℜ decomposes as X⊗Y where Y=M2. Let I=M°1⊖m, so M=N⊗Y where N2=M°1⊖m2=M2°1⊖m2=m°1⊖m=0. To show N is weakly prime hyperideal, let a°b∈N2−0; so a,1°b,1=a°b,1∈N⊗Y−N⊗Y2=N⊗Y−0⊗Y⊆M−ϕM. Since ϕ≤ϕ3, then it follows ϕM⊆M3=N⊗Y3=0⊗Y. We obtain that a,1∈M or b,1∈M. Therefore a∈N or b∈N. As a consequence N is weakly prime hyperideal of ℜ.Proposition 3.
Letℜ1andℜ2be commutative Krasner hyperrings and letφi:Lℜi⟶Lℜi∪∅be a function fori=1,2. Takeϕ=φ1×φ2andℜ=ℜ1×ℜ2. ThenNisϕ-prime hyperideal ofℜif and only ifNis one of the following types:(i)
N=N1×N2,whereNiis a proper hyperideal ofℜiwithφiNi=Ni.(ii)
N=N1×ℜ2,whereN1isφ1-prime hyperideal ofℜ1that should be prime ifφ2ℜ2≠ℜ2.(iii)
N=ℜ1×N2,whereN2isφ2-prime hyperideal ofℜ2that should be prime ifφ1ℜ1≠ℜ1.Proof.
⇒ (i) Obviously N is ϕ-prime hyperideal, since N1×N2−ϕN1×N2=∅(ii) LetN1 is φ1-prime hyperideal of ℜ1 and φ2ℜ2≠ℜ2. Assume that a1°b1,a2°b2=a1,a2°b1,b2∈N1×ℜ2−φ1N1×φ2ℜ2=N1−φ1N1×ℜ2−φ2ℜ2 for a1,a2,b1,b2∈ℜ=ℜ1×ℜ2. Thus a1°b1∈N1−φ1N1 and then either a1∈N1 or b1∈N1. So a1,a2∈N1×ℜ2 or b1,b2∈N1×ℜ2. Therefore N1×ℜ2 is ϕ-prime hyperideal of ℜ.(iii) The proof is similar to (ii).⇐ Assume that N is a ϕ-prime hyperideal of ℜ where φiNi≠Ni. Let a°b∈N1−φ1N1 for some a,b∈ℜ1. So a,0°b,0=a°b,0∈N−ϕN. Since N is a ϕ-prime hyperideal of ℜ, then a,0∈N or b,0∈N. So a∈N1 or b∈N1. Therefore N1 is φ1-prime hyperideal of ℜ1. Similarly we can find N2 is φ2-prime hyperideal of ℜ2. Now we need to show that N1=ℜ1 or N2=ℜ2. Suppose that N2≠ℜ2. Take b1∈N1−φ1N1, b2∈ℜ2−N2. Then note that 1,0°b1,b2=b1,0∈N−ϕN. Since N2≠ℜ2, then 1,0∈N and so we find 1∈N1. We get N1=ℜ1. Similarly one can easily find N2=ℜ2, if N1≠ℜ1. Without loss of generality N1≠ℜ1. Now let we show that N1 is prime hyperideal with φ2ℜ2≠ℜ2. For m′∈ℜ2−φ2ℜ2, let x°m∈N1 for some x,m∈ℜ1. Then we conclude that x,1°m,m′=x°m,m′∈N−ϕN. Since N is a ϕ-prime hyperideal of ℜ, then we find x,1∈N or m,m′∈N which implies that x∈N1 or m∈N1. Therefore N1 is a prime hyperideal of ℜ1. If φ1ℜ1≠ℜ1 and N1=ℜ1, then similarly one can prove that N2 is a prime hyperideal of ℜ2.


## 3. Generalizations of Primary Hyperideals in Krasner Hyperrings

Similar to the previous section, we considerℜ,⊕,° to be a commutative Krasner hyperring with nonzero unit. We denote the set of all hyperideals of ℜ by Lℜ. Let we define ϕ-primary hyperideal.Definition 3.
Letℜbe a commutative hyperring andNbe a proper hyperideal ofℜ.Letϕbe a function such thatϕ:Lℜ⟶Lℜ∪∅.Nis calledϕ-primary hyperideal ofℜifa°b∈N−ϕN, then eithera∈Norbk∈Nfor somea,b∈ℜ,k∈ℕ.Example 2.
Letℜbe a commutative Krasner hyperring. Then we define functionsϕ:Lℜ⟶Lℜ∪∅such as in Example 1. Also we have the same order,ϕ⊘≤ϕ0≤ϕw≤…≤ϕn+1≤ϕn≤ϕn−1≤…≤ϕ2≤ϕ1.Definition 4.
Letℜbe a hyperring andNbe a proper hyperideal ofℜ.(i)
Nis primary hyperideal if and only if it isϕ∅-primary hyperideal.(ii)
Nis weakly primary hyperideal if and only if it isϕ0-primary hyperideal.(iii)
Nis almost primary hyperideal if and only if it isϕ2-primary hyperideal.(iv)
Nisn-almost primary hyperideal if and only if it isϕn-primary hyperideal.(v)
Nisw-primary hyperideal if and only if it isϕw-primary hyperideal.Proposition 4.
LetNbe a proper hyperideal ofℜ.(1)
Ifσ1,σ2are two functions withσ1≤σ2such thatσ1,σ2:Lℜ⟶Lℜ∪∅andNisσ1-primary, thenNisσ2-primary.(2)
(i)Nis primary hyperideal⇒Nis weakly primary hyperideal⇒Nisw-primary hyperideal⇒Nisn+1-almost primary hyperideal⇒Nisn-almost primary hyperideal, forn≥2⇒Nis almost primary hyperideal.(ii)N is w-primary hyperideal if and only if N is n-almost primary hyperideal for all n≥2.Proof.
(1)
We supposeN is a σ1-primary hyperideal of ℜ. Take a°b∈N−σ2N for a,b∈ℜ. It means a°b∈N−σ1N. Because of N is σ1-primary hyperideal of ℜ , a∈N or bk∈N, for some k∈ℕ. Therefore N is σ2-primary hyperideal.(2)
(i) We can obtain it by the ordering of theϕi′s given in 2. (ii) Assume that a°b∈N−Nn for all n≥2. Then a°b∈N−∩n=2∞Nn and we find a°b∈N−∩n=1∞Nn. Since N is w-primary hyperideal of ℜ, then we have a∈N or bk∈N, for some k∈ℕ . Conversely, if a°b∈N−∩n=1∞Nn, then a°b∈N−Nn for some n≥1. Actually a°b∈N−Nn for some n≥2. Therefore a∈N or bk∈N for some k∈ℕ, since N is n-almost primary for all n≥2.
The next theorem shows us how to determineϕ-primary hyperideal to be primary. We call it as a characterization.Theorem 4.
Letϕ:Lℜ⟶Lℜ∪∅be a function andTbe a proper hyperideal ofℜ, such thatTis aϕ-primary hyperideal ofℜ. IfTis not primary, thenT2⊆ϕT. Hence at the same time it means ifT2⊈ϕT, thenTis primary hyperideal ofℜ.Proof.
Assume thatT2⊈ϕT. We need to see that T is primary. Take some x°y∈T for x,y∈ℜ. If x°y∉ϕT, then since T is ϕ-primary, x∈T or yk∈T for some k∈ℕ. If x°y∈ϕT, assuming that x°T⊈ϕT, we get x°p0∉ϕT, where p0∈T. We have x°y⊕p0⊆T−ϕT. Thus x∈T or y⊕p0k⊆T for some k∈ℕ. It follows x∈T or yk∈T. Now we can assume that x°T⊆ϕT. (In the same way we can assume that y°T⊆ϕT). Because of T2⊈ϕT, there exist p1,q1∈T with p1°q1∉ϕT. Then x⊕p1∘y⊕q1⊆T−ϕT. As T is ϕ-primary, so x⊕p1⊆T or y⊕q1m⊆T for some m∈ℕ. Consequently T is primary hyperideal of ℜ.Corollary 3.
IfTis aϕ-primary hyperideal ofℜ, whereϕ≤ϕ3, thenTisw-primary hyperideal.Proof.
The proof is similar to Corollary1.
In the following, some characterizations ofϕ-primary hyperideals are provided.Theorem 5.
LetNbe a proper hyperideal of the commutative Krasner hyperringℜand letϕ:Lℜ⟶Lℜ∪∅be a function. The following statements hold:(i)
Nisϕ-primary hyperideal ofℜ.(ii)
Fora∈ℜ−N;N:a=N∪ϕN:a.(iii)
Fora∈ℜ−N;N:a=NorN:a=ϕN:a.(iv)
For each hyperidealsKandLofℜ, ifK∘L⊆NandK∘L⊈ϕN, thenK⊆NorL⊆NProof.
i⇒ii Let N is ϕ-primary hyperideal of ℜ. It is obvious that N∪ϕN:a⊆N:a. To prove the other side; for every b∈N:a, we have a∘b∈N. If a∘b∈N−ϕN, then b∈N since N is ϕ-primary and a∈ℜ−N. If a∘b∈ϕN then b∈ϕN:a. So N:a⊆N∪ϕN:a. Hence N:a=N∪ϕN:a.ii⇒iii It is obvious because of N:a is a hyperideal of ℜ.iii⇒iv Let K and L be hyperideals of ℜ, with K∘L⊆N−ϕN. Assume that L⊆N. Then there exists an element a∈L−N by iii we have N:a=N or N:a=ϕN:a. If K∘a⊈ϕN, then K⊈ϕN:a. Since iii holds, then K⊆N:a=N. We are done. Assume that K∘a⊆ϕN. Since K∘L⊆ϕN, then we can choose an element x∈L such that K∘x⊆N−ϕN. If x∉N, then by iii, N:x=N or N:x=ϕN:x. Since K⊆N:x, but K⊆ϕN:x, then we conclude that K⊆N:x=N. So assume that x∈N. Then we have a⊕x⊆L−N, and also note that K∘a⊕x⊆N−ϕN, since K∘a⊆ϕN and K∘x⊈ϕN. Then K⊆N:a⊕x=N which completes the proof.iv⇒i Let a∘b∈N−ϕN. Then a∘b⊆N, but a∘b⊈ϕN. So a⊆N or b⊆N. Therefore a∈N or bm∈N for some m∈ℕ.
In Theorem4, we prove that if T is a ϕ-primary hyperideal that is not primary, then T2⊆ϕT.Proposition 5.
Letϕ:Lℜ⟶Lℜ∪∅be a function andTbe aϕ-primary hyperideal ofℜ.(i)
IfTis an hyperideal ofℜwithM⊆T, thenT/MisϕM-primary hyperideal ofℜ/M.(ii)
LetSbe a multiplicatively closed subset ofℜwithT∩S=∅andϕTS⊆ϕTS. ThenTSis anϕS-primary hyperideal ofℜS.Proof.
(i)
Letx,y∈ℜ. We assume that x⊕M∘y⊕M⊆T/M−ϕMT/M, it means x∘y⊕M⊆T/M−ϕT⊕M/M. We find x∘y∈T−ϕT⊕M. Then x∘y∈T−ϕT and so x∈T or yk∈T for some k∈ℕ. Hence x⊕M⊆T/M or y⊕Mk⊆T/M. This gives us T/M is ϕM-primary hyperideal of ℜ/M.(ii)
Leta/s∘b/t∈TS−ϕSTS, for some a,b∈ℜ;s,t∈S. Then we have p∘a∘b∈T for some p∈S but q∘a∘b∉ϕT∩ℜ for every q∈S. Now if q∘a∘b∈ϕT, then a/s∘b/t∈ϕTS⊆ϕSTS gives us a contradiction. So p∘a∘b∈T−ϕT and since T is ϕ-primary, then we get either p∘a∈T or bk∈T for some k∈ℕ. Therefore a/s∈TS or bk/tk∈TS because of T∩S=∅ it is same with T∩S=∅). Thus TS is ϕS-primary hyperideal of ℜS.Theorem 6.
(i)
LetX,Ybe commutative Krasner hyperrings andNbe a weakly primary hyperideal ofX. ThenM=N⊗Yis aϕ-primary hyperideal ofℜ=X⊗Y, for allϕwithϕw≤ϕ≤ϕ1.(ii)
Letℜbe a commutative Krasner hyperring andMbe a finitely generated proper hyperideal ofℜ, such thatMis aϕ-primary hyperideal withϕ≤ϕ3. ThenMis either weakly primary orM2≠0is idempotent andℜdecomposes asX⊗Y, whereY=M2andM=N⊗Y, whereNis weakly primary. As a resultMisϕ-primary for eachϕwithϕw≤ϕ≤ϕ1.Proof.
(i)
LetN be a weakly primary hyperideal of X. M is weakly primary if and only if M is primary. Nevertheless Mis a ϕ-primary hyperideal for each ϕ with ϕw≤ϕ. If N is primary, then M is primary and hence ϕ-primary for all ϕ. Assume that N is not primary. Then N2=0. Therefore M2=0⊗Y and hereby ϕwM=0⊗Y. Hence we can write M−ϕwM in this form M−ϕwM=N⊗Y−0⊗Y=N−0⊗Y. Let we take a1,a2°b1,b2=a1°b1,a2°b2∈M−ϕwM. It means a1°b1∈N−0, so a1∈N or b1s∈N for some s∈ℕ. Then a1,a2∈N⊗Y or b1,b2s∈N⊗Y. Hence M=N⊗Yis ϕw-primary, therefore M is ϕ -primary hyperideal.(ii)
IfM is primary hyperideal, then M is weakly primary hyperideal of ℜ. Suppose that M is not primary. From Theorem 4, M2⊆ϕM and hence M2⊆ϕMϕ3M=M3. So M2=M3, which means M2 is idempotent. Since M2 is finitely generated, then M2=m for some idempotent m∈ℜ. Assume that M2=0. Then ϕM⊆M3=0. Hereby ϕM=0. Consequently, M is weakly primary hyperideal of ℜ. Now let us suppose M2≠0. Take Y=M2=ℜ∘m and X=ℜ∘1⊖m, so ℜ decomposes as X⊗Y, where Y=M2. Let X=M∘1⊖m, so M=N⊗Y, where N2=M∘1⊖m2=M2∘1⊖m2=m∘1⊖m=0. To show N is weakly primary hyperideal, let a∘b∈N2−0. Thus a,1∘b,1=a∘b,1∈N⊗Y−N⊗Y2=N⊗Y−0⊗Y⊆M−ϕM, since ϕ≤ϕ3 implies ϕM⊆M3=N⊗Y3=0⊗Y. We obtain that a,1∈M or b,1t∈M, for some t∈ℕ. Therefore a∈N or bt∈N. As a consequence, N is weakly primary hyperideal of ℜ.Proposition 6.
Letℜ1andℜ2be commutative Krasner hyperrings and letφi:Lℜi⟶Lℜi∪∅be a function, fori=1,2. Takeφ=φ1×φ2andℜ=ℜ1×ℜ2. ThenNisϕ-primary hyperideal ofℜif and only ifNis one of the following types:(i)
N=N1×N2whereNiis a proper hyperideal ofℜiwithφiNi=Ni.(ii)
N=N1×ℜ2whereN1isφ1-primary hyperideal ofℜ1that should be primary ifφ2ℜ2≠ℜ2.(iii)
N=ℜ1×N2whereN2isφ2-primary hyperideal ofℜ2that should be primary ifφ1ℜ1≠ℜ1.Proof.
⇒ (i) Obviously N is ϕ-primary hyperideal, since N1×N2−ϕN1×N2=∅.(ii) LetN1 be φ1-primary hyperideal of ℜ1 and φ2ℜ2≠ℜ2. Assume that a1∘b1,a2∘b2=a1,a2∘b1,b2∈N1×ℜ2−φ1N1×φ2ℜ2=N1−φ1N1×ℜ2−φ2ℜ2, for a1,a2,b1,b2∈ℜ=ℜ1×ℜ2. We have a1∘b1∈N1−φ1N1, then either a1∈N1 or b1k∈N1, for some k∈ℕ, since N1 is φ1-primary hyperideal of ℜ1. So a1,a2∈N1×ℜ2 or b1k,b2k=b1,b2∈N1×ℜ2. Therefore N1×ℜ2 is ϕ-primary hyperideal of ℜ.(iii) The proof is similar to (ii).⇐ Suppose that N is a ϕ-primary hyperideal of ℜ, where φiNi≠Ni. Let a∘b∈N1−φ1N1, for some a,b∈ℜ1. Thus a,0∘b,0=a∘b,0∈N−ϕN. Since N is a ϕ-primary hyperideal of ℜ, then a,0∈N or b,0k∈N for some K∈ℕ. So a∈N1 or bk∈N1. Therefore N1 is φ1 -primary hyperideal of ℜ1. Similarly we can find N2 is φ2-primary hyperideal of ℜ2. We have to show now that N1=ℜ1 or N2=ℜ2. Suppose that N2≠ℜ2. Let take b1∈N1−φ1N1, b2∈ℜ2−N2. Then note that 1,0∘b1,b2=b1,0∈N−ϕN. Since N2≠ℜ2, 1,0t=1,0∈N (N2≠ℜ2), then we find 1∈N1 and so we get N1=ℜ1. Similarly one can easily find N2=ℜ2, if N1≠ℜ1. Without loss of generality N1≠ℜ1. Now let we show that N1 is primary hyperideal with φ2ℜ2≠ℜ2. For m′∈ℜ2−φ2ℜ2, let x∘m∈N1, for some x,m∈ℜ1 . Then we conclude that x,1∘m,m′=x∘m,m′∈N−ϕN. Since N is a ϕ-primary hyperideal of ℜ, then we find x,1S∈N for some s∈ℕ or m,m′∈N which implies that xs∈N1 or m∈N1. Therefore N1 is a primary hyperideal of ℜ1. If φ1ℜ1≠ℜ1 and N1=ℜ1, then similarly one can prove that N2 is a primary hyperideal of ℜ2.


## 4.ϕ-δ-Primary Hyperideals in Krasner Hyperrings

LetN be a proper hyperideal of hyperring ℜ. Denote the set of all hyperideals of ℜ, by Lℜ and denote the set of all proper hyperideals of ℜ, by L∗ℜ. The function ϕ:Lℜ⟶Lℜ∪∅ is said to be reduction function if ϕN⊆N and N⊆M implies ϕN⊆ϕM, for each N,M∈Lℜ and δ be an expansion function such that δ:Lℜ⟶Lℜ. Now, we give some examples related to reduction and expansion functions.Example 3.
Letℜbe a commutative Krasner hyperring with a nonzero identity. Let us consider the following functionsδonLℜ, for anyN∈Lℜ,(i)
δ0N=N, i.e.,δis the identity function.(ii)
δ1N=N, i.e.,δis the radical operation.(iii)
δresN=N:Mfor a fixedM∈Lℜ.(iv)
δannN=annannN.(vi)
δMN=N⊕Mfor a fixedM∈Lℜ.
All the above functions are examples of expansion onLℜ.Example 4.
Letℜbe a commutative Krasner hyperring with a nonzero identity. Consider the following functionsϕ:Lℜ⟶Lℜ∪∅defined as follows: for anyN∈Lℜ:(i)
ϕ∅N=∅.(ii)
ϕ∅N=0.(iii)
ϕ1N=N.(iv)
ϕ2N=N2.(v)
ϕkN=Nk.(vi)
ϕkN=∩i=1∞Ni.
All the above functions are reduction onLℜ. Remember that ϕ∅≤ϕ0≤ϕw≤…≤ϕn+1≤ϕn≤ϕn+1≤…≤ϕ2≤ϕ1.Definition 5.
Letδbe a hyperideal expansion,ϕbe a hyperideal reduction andNbe a proper hyperideal ofℜ.Nis said to be aϕ-δ-primary hyperideal ifa∘b∈N−ϕN, then eithera∈Norb∈δNfor eacha,b∈ℜ.Remark 1.
[24] Ifδ1,δ2,…,δnare hyperideal expansions, thenδ=∩i=1nδiis also an hyperideal expansion.Definition 6.
Letℜbe a hyperring andNbe a proper hyperideal ofℜ(i)
Nis prime hyperideal if and only if it isϕ∅-δ0-primary hyperideal [24].(ii)
Nis primary hyperideal if and only if it isϕ∅-δ1-primary hyperideal [24].(iii)
Nisϕ-prime hyperideal if and only if it isφ-δ0-primary hyperideal.(iv)
Nisϕ-primary hyperideal if and only if it isφ-δ1-primary.(v)
Nisδ-primary hyperideal if and only if it isφ∅-δ-primary hyperideal.(vi)
Nis weakly prime hyperideal if and only if it isϕ0-δ0-primary hyperideal.(vii)
Nis almost prime hyperideal if and only if it isφ2-δ0-primary hyperideal.
Suppose thatϕ,σ are reductions on Lℜ. Then we write φ≤σ if ϕN⊆σN, for all N∈Lℜ. Similarly, for any two expansions δ,γ on Lℜ;δ≤γ if δN⊆γN, for each N∈Lℜ.Definition 7.
Letℜbe a hyperring andNbe a proper hyperideal ofℜ.(i)
IfNisϕ0-δ-primary hyperideal, thenNis said to be a weaklyδ-primary hyperideal ofℜ.(ii)
IfNisϕ2-δ-primary hyperideal, thenNis said to be an almostδ-primary hyperideal ofℜ.(iii)
IfNisϕn-δ-primary hyperideal, thenNis said to be anr1-almostδ-primary hyperideal ofℜ.(iv)
IfNisϕw-δ-primary hyperideal, thenNis said to be aw-δ-primary hyperideal ofℜ.Proposition 7.
Letℜbe a hyperring,Nbe a proper hyperideal ofℜ,ϕ,φbe reductions onLℜandδ,γbe expansions onLℜ. The following statements hold:(i)
Ifϕ≤σ, then everyϕ-δ-primary hyperideal is aσ-δ-primary hyperideal.(ii)
Ifδ≤γ, then everyϕ-δ-primary hyperideal is aϕ-γ-primary hyperideal.(iii)
Everyϕ-prime hyperideal is aϕ-δ-primary hyperideal.(iv)
Everyδ-primary hyperideal is aϕ-δ-primary hyperideal.(v)
Suppose thatNis proper hyperideal ofℜ.Nis weaklyδ-primary⇒Nisw-δ-primary hyperideal⇒Nisr1-almostδ-primary, for eachn≥2⇒Nis almostδ-primary.Proof.
iii: Straightforward.iii: It follows from (ii) and Definition 6 (iii), since δ0≤δ.iv: It follows from (i) and Definition 6 (iii), since ϕ∅≤ϕ.v: It follows from (i) and the fact that ϕ∅≤ϕw≤ϕn≤ϕ2.Proposition 8.
Letϕbe a hyperideal reduction,δbe a hyperideal expansion andmi:i∈Δbe a directed family ofϕ-δ-primary hyperideals ofℜ. ThenM=∪ieΔMiis aϕ-δ-primary hyperideal.Proof.
Letmi:i∈Δ be a directed family of ϕ-δ -primary hyperideals of ℜ. Assume that a∘m∈M−ϕ∪ieΔMi. This implies that a∘m∈Mi−ϕMi, for some i∈Δ. We get either a∘m∈Mi or m∈δMi, because of M1 is a ϕ-δ-primary. If a∈Mi, then clearly we have a∈∪ieΔMi. If m∈δMi, then we have m∈δ∪ieΔMi, since M⊆∪ieΔMi. Hence M=∪ieΔMi is a ϕ-δ-primary hyperideal.
In the following, we give a characterization forϕ-δ-primary hyperideals such that ϕ is a hyperideal reduction and δ is a hyperideal expansion.Theorem 7.
Letℜbe a Krasner hyperring andNbe a proper hyperideal ofℜ. Then the following statements hold:(i)
Nis aϕ-δ-primary hyperideal;(ii)
For eacha∈ℜ−δN,N:a=N∪ϕN:a;(iii)
For eacha∈ℜ−δN,N:a=NorN:a=ϕN:a;(iv)
For each hyperidealK,Lofℜ,K∘L⊆NandK∘L⊆ϕNimply thatK⊆NorL⊆δN.(v)
For each hyperidealMofℜsuch thatM⊈δN, thenN:M=NorN:M=ϕN:M.Proof.
i⇒ii Suppose that N is a ϕ-δ-primary hyperideal and a∈ℜ−δN. It is clear that N∪ϕN:a⊆N:a. Let m∈N:a. Then we have a∘m∈N. If a∘m∈ϕN, then we obtain m∈ϕN:a⊆N∪ϕN:a. Assume that a∘m∉ϕN. Since a∘mN−ϕN and a∉δN, then we get m∈N⊆N∪ϕN:a. Hence, N:a=N∪ϕN:a.ii⇒iii It follows from the fact that a hyperideal is a union of two hyperideals. Then it must be equal to one of them.iii⇒iv: Let K∘L⊆N, for some hyperideals K and L of ℜ. Assume that L⊈δN and K⊈N. Then there exists m∈L−δN. We have to show K∘L⊆ϕN. By (iii), we have either N:m=N or N:m=ϕN:m. If K∘m⊆N, then by (iii), we get K⊆N:m=N For a∈K−N, it means a∈N:m−N. Hence by part (iii), N:m=ϕN:m. Thus K⊆N:m=ϕN:m implies that K∘m⊆ϕN. On the other side, suppose that m∈δN. Then m∈L∪δN. Choose an element m′∈L−δN so m⊕m′⊆L−δN. Hence K∘m′⊆ϕN and K∘m⊕m′⊆ϕN. Let a∈K. Then am⊕m′⊖a∘m′=a∘m′⊆ϕN. Thus K∘m⊆ϕN. Therefore K∘L⊆ϕN.iv⇒v: Suppose that M is a hyperideal of ℜ such that M⊈δN. Also, note that M∘N:M⊆N. If M∘N:M⊆ϕN, then we have N:M⊆ϕN:M⊆N:M. Assume that M∘N:M⊈ϕN. Then by (iv), N:M⊆N⊆N:M.v⇒i: Suppose that a∘m∈N−ϕN, with a∉δN, for a,m∈ℜ. Put ℜ∘a=M and note that m∈N:M. Then by (v), we get either m∈N:M=N or m∈N:M=ϕN:M. The latter case is impossible, since a∘m∉ϕN. Therefore, N is a ϕ-δ-primary hyperideal.Theorem 8.
(i)
LetTbe aϕ-δ-primary hyperideal ofℜsuch thatϕT:a=ϕT:a, for eacha∈ℜ. ThenT:ais aϕ-δ-primary hyperideal ofℜ.(ii)
Suppose thatδ≤δ1are hyperideal expansions andTis aϕ-δ-primary hyperideal ofℜsuch thatδ1ϕT⊆δT. ThenδT=δ1T.Proof.
i Let T be a ϕ-δ-primary hyperideal of ℜ such that ϕT:a=ϕT:a, for each a∈ℜ. We show that T:a is a ϕ- δ-primary hyperideal of ℜ. For this, let we take x,m∈ℜ such that x∘m∈T:a−ϕT:a. Then we have x∘a∘m∈T. Since ϕT:a=ϕT:a, then we also have x∘a∘m∉ϕT. Since T is a ϕ-δ-primary hyperideal, then we get either x∈δT or a∘m∈T implying either x∈δT:a or m∈T:a. Therefore, T:a is a ϕ-δ -primary hyperideal of ℜ.ii Since δ≤δ1 then we have δT⊆T=δ1T. For the converse, take x∈T. Then there exists minimal k∈ℕ such that xk∈T, that is, xk−1∉T. If K=1, then we have x∈T⊆δT. Now, assume that K>1. We have two cases. Case 1: Let x∈δ1ϕT=ϕT. Then we have x∈δT. Case 2: Let x∉ϕT. Then we have xk∉ϕT, that is, xk−1∘x∘ℜ⊆T and x∘xk−1∘ℜ⊈ϕT. Since T is a ϕ-δ -primary hyperideal of ℜ, then by Theorem 7, x∈δT or xK−1∘ℜ⊆T. The latter case is impossible, since xK−1∉T. Thus in both cases, we have x∈δT. Therefore, δT=δ1T.Theorem 9.
LetTbe aϕ-δ-primary hyperideal ofℜsuch thatδT∘T⊆ϕT. ThenTis aδ-primary hyperideal ofℜ.Proof.
Leta∘m∈T, for some a,m∈ℜ. If a∘m∉ϕT, then we conclude either a∈δT or m∈T as T is an ϕ-δ -primary hyperideal of ℜ. Assume that a∘m∈ϕT. If a∘T⊈ϕT, then there exists n∈T such that a∘n∉ϕT. Thus we have a∘m⊕n⊆T−ϕT, which implies either a∈δT or m⊕n⊆T. Then we get a∈δT or m∈T, which completes the proof. Assume that a∘T⊆ϕT. Similarly, we may assume that δT∘m⊆ϕT . As δT∘T⊆ϕT, we can find b∈δT and m′∈T such that b∘m′⊆ϕT. Then we conclude that a⊕b∘m⊕m′⊆T−ϕT . Since T is a ϕ-δ-primary hyperideal of ℜ, then we have either a⊕b⊆δT or m⊕m⊆T, which implies that a∈δT or m∈T. Therefore, T is a δ-primary hyperideal of ℜ.Definition 8.
(i)
[24] A hyperideal expansionδis said to be global if for any hyperring good homomorphismμ:ℜ⟶S^, δμ−1M=μ−1δM, for eachM∈LS^.(ii)
A hyperideal reductionϕis said to be a global if for any homomorphismμ:ℜ⟶S^,ϕμ−1M=μ−1δM, for eachM∈LS^.
For instance, the hyperideal reductionsϕ0,ϕ1 and the hyperideal expansions δ0,δ1 are both global.Theorem 10.
Letμbe a good homomorphism from Krasner hyperringℜ,⊕,∘into a Krasner hyperringS^,+,·. The following statements hold:(i)
Letμ:ℜ⟶S^be a good homomorphism andMbe aϕ-δ-primary hyperideal ofS^such thatϕis global reduction function andδis global expansion function. Thenμ−1M=ℜorμ−1Mis aϕ-δ-primary hyperideal ofℜ.(ii)
Letμ:ℜ⟶S^be a good epimorphism andNbe a hyperideal ofℜcontainingKerμ. Suppose thatϕis global reduction function andδis global expansion function. ThenNis aϕ-δ-primary hyperideal ofℜif and only ifμNis aϕ-δ-primary hyperideal ofS^.Proof.
i Let μ:ℜ⟶S^ be a hyperring homomorphism and M be a ϕ-δ-primary hyperideal of S^ such that μ−1M≠ℜ. Assume a∘m∈μ−1M−ϕμ−1M, for some a,m∈ℜ. Since ϕ is global, then ϕμ−1M=μ−1ϕM and thus a∘m∈μ−1M−μ−1ϕM. This implies that μa∘m=μa⋅μm∈M=ϕM. Since M is a ϕ-δ-primary hyperideal of S^, then we have either μa∈δM or μm∈M. Since M⊆μ−1M, then we get a∈μ−1δM=δμ−1M or M⊆μ−1M. Therefore, μ−1M is a ϕ-δ-primary hyperideal.ii Let us suppose that μN is a ϕ-δ-primary hyperideal of S^, where N is a hyperideal of ℜ containing Kerμ. By i, μ−1μN=N is a ϕ-δ-primary hyperideal of ℜ. For the converse, let N be a ϕ-δ-primary hyperideal of ℜ and a′⋅m′∈μN−ϕμN, for some a′,m′∈S^. Then ∃a,m∈ℜ with μa=a′ and μm=m′, since μ is surjective. a′⋅m′=μa⋅μm=μa°m=μx∈μN−ϕμN, for some x∈ℜ. So 0∈μa°m−μx=μa°m⊖x. Hence there exists t∈a°m⊖x such that μt=0S^. a°m∈t⊕x⊆Kerμ+N⊆N+N⊆N. Then ϕ is global and Kerμ⊆N, we have ϕμN=μϕN and also a°m∈N−ϕN. Since N is a ϕ-δ-primary hyperideal of ℜ, then we have either a∈δN or m∈N. Since N⊆μN, then we have a′=μa∈δμN or m′=μm∈μN. Therefore, μN is a ϕ-δ-primary hyperideal of S^.
As an instant consequence of the previous theorem, we get the following explicit results.Corollary 4.
(i)
LetNbe aϕ-δ-primary hyperideal ofℜandMbe a hyperideal ofℜwithM⊈N. Suppose thatϕis global reduction function andδis global expansion function. ThenN∩Mis aϕ-δ-primary hyperideal ofM.(ii)
LetNandMbe two hyperideals ofℜwithM⊆N. Suppose thatϕis global reduction function andδis a global expansion function. ThenNis aϕ-δ-primary hyperideal ofℜif and only ifN/Mis aϕ-δ-primary hyperideal ofℜ/M.Theorem 11.
LetNbe a hyperideal ofℜ. ThenNis aϕ-δ-primary hyperideal ofℜif and only ifN/ϕNis a weaklyδ-primary hyperideal ofℜ/ϕN.Proof.
Assume thatN is a ϕ-δ-primary hyperideal of ℜ. Let a⊕ϕN°m⊕ϕN⊆N/ϕN−0ℜ/ϕN. Then we have a°m∈N−ϕN. Since N is a ϕ-δ-primary hyperideal of ℜ, then we get a∈δN or m∈N, which implies a⊕ϕN∈δN/ϕN or m⊕ϕN⊆N/ϕN. Hence, N/ϕN is a weakly δ-primary hyperideal of ℜ/ϕN. For the converse, let a,m∈ℜ such that a°m∈N−ϕN. This implies that a⊕ϕN°m⊕ϕN⊆N/ϕN−0ℜ/ϕN. Since N/ϕN is a weakly δ-primary hyperideal of ℜ/ϕN, then we get a⊕ϕN∈δN/ϕN or m⊕ϕN⊆N/ϕN implying a∈δN or m∈N. Therefore, N is a ϕ-δ -primary hyperideal of ℜ.
Letℜ be a commutative Krasner hyperring, S⊆ℜ be a multiplicatively closed subset of ℜ and T be a proper hyperideal of ℜ.Proposition 9.
LetϕS:LℜS⟶LℜS∪∅be a hyperideal reduction function andδS:LℜS⟶LℜSbe a hyperideal expansion function such thatδSNS=δNS, for eachN∈Lℜ. IfTis aϕ-δ-primary hyperideal such thatT∩S=∅andϕTS⊆ϕSTS, thenTSis aϕS-δS-primary ofℜS.Proof.
Leta/s°m/t∈TS−ϕSTS, for some a,m∈ℜ;s,t∈S. Then we have p°a°m∈T−ϕT, for some p∈S, since ϕTS⊆ϕSTS. Since T is a ϕ-δ-primary hyperideal of ℜ, then we have either a∈δT or p°m∈T. If a∈δT, then a/s∈δTS=δSTS. If p°m∈T, then we have m/t=p°m/p°t∈TS. Therefore, TS is a ϕS-δS-primary hyperideal of ℜS.
Letφi:Lℜi⟶Lℜi∪∅ be hyperideal reduction functions and γi:Lℜi⟶Lℜi be hyperideal expansion functions for i=1,2. Suppose that ℜ=ℜ1×ℜ2. Also, each hyperideal N of ℜ has the form N=N1×N2, where Ni is a hyperideal of ℜi. Furthermore, ϕ:Lℜ⟶Lℜ∪∅ defined by ϕN1×N2=φ1N1×φ2N2 is a hyperideal reduction function and also the function δ:Lℜ⟶Lℜ defined by δN1×N2=γ1N1×γ2N2 is a hyperideal expansion function.Proposition 10.
Let the notation be as in the above pharagraph andN=N1×N2. Then each of the types ofNareϕ-δ-primary hyperideals ofℜ=ℜ1×ℜ2.(i)
N=N1×N2, whereNiis a proper hyperideal ofℜiwithφiNi=Ni.(ii)
N=N1×ℜ2, whereN1isγ1-primary hyperideal ofℜ1.(iii)
N=ℜ1×N2, whereN2isγ2-primary hyperideal ofℜ2.(iv)
N=N1×ℜ2, whereN1isφ1-γ1-primary hyperideal ofℜ1andφ2ℜ2=ℜ2.(v)
N=ℜ1×N2, whereN2isφ2-γ2-primary hyperideal ofℜ2andφ1ℜ1=ℜ1..Proof.
(i)
ObviouslyN is ϕ-δ-primary hyperideal, since N1×N2−ϕN1×N2=⊘.(ii)
LetN=N1×ℜ2, where N1is γ1-primary hyperideal of ℜ1. We can see that N1×ℜ2 is δ -primary hyperideal of ℜ1. By the Proposition 7, N is ϕ -δ-primary hyperideal of ℜ.(iii)
The proof is similar to (ii).(iv)
Suppose thata1,a2°m1,m2∈N1×ℜ2−ϕN1×ℜ2=φ1N1×φ2ℜ2. Then a1°m1,a2°m2∈N1−φ1N1×ℜ2−φ2ℜ2. Since N1is φ1-γ1-primary hyperideal of ℜ1, then we get either a1∈N1 or m1∈γ1N1. So a1,a2∈N1×ℜ2 or m1,m2∈γ1N1×ℜ2⊆γ1N1×γ2ℜ2. It means N is a ϕ-δ-primary hyperideal of ℜ.(v)
The proof is similar to (iv).Theorem 12.
Let the notation be as in the Proposition 10,N=N1×N2, whereφ2Ni≠Ni. ThenNis aφ, δprimary hyperideal ofℜif and only ifNis one of the following types:(i)
N=N1×ℜ2, whereN1isφ1, γ1-primary hyperideal ofℜ1that should beγ1-primary ifφ2ℜ2≠ℜ2.(ii)
N=ℜ1×N2, whereN2isφ2, γ2-primary hyperideal ofℜ2that should beγ2-primary ifφ1ℜ1≠ℜ1.Proof.
⇐ It follows from the Proposition 10.⇒ Suppose that N is a φ, δ-primary hyperideal of ℜ, where φiNi≠Ni. Let a∘m∈N1−φ1N1, for some a,m∈ℜ1. Thus a,0∘m,0=a∘m,0∈N−φN. Since N is a φ-δ -primary hyperideal of ℜ, then a,0∈N or m,0∈δN. So a∈N1 or m∈γ1N1. Therefore N1 is φ1, γ1-primary hyperideal of ℜ1. Similarly, we can find N2 is φ2-γ2-primary hyperideal of ℜ2. We have to show now that N1=ℜ1 or N2=ℜ2. Suppose that N2≠ℜ2. Let we take m1∈N1−φ1N1, m2∈−ℜ2−N2. Notice that 1,0∘δm1,m2=m1,0∈N−φN. This implies that 1,0∈δN, so we find 1∈γ1N1. Then N1=ℜ1. Similarly, one can easily find N2=ℜ2, if N2≠ℜ2. Without loss of generality N1≠ℜ1. Let we show now that N1 is γ1 primary hyperideal with φ2ℜ2≠ℜ. For m∈ℜ2−φ2ℜ2, let x∘m∈N, for some x,m∈ℜ1. We have that x,1∘m,m′=x∘m,m′∈N−φN. Since N is a φ-δ primary hyperideal ofthen we find x,1∈δN=γ1N1×γ2N2 or m,m′∈N which implies that x∈γ1N1 or m∈N1. Therefore N1 is a γ1-primary hyperideal of ℜ1 If ϕℜ1≠ℜ1 and N1≠ℜ1, then similarly one can prove that N2 is a γ2 primary hyperideal of ℜ2.


## 5. Conclusion

In this paper, generalizations of prime and primary hyperideals were provided using the functionϕ. We introduced ϕ-prime, ϕ primary and ϕ-δ primary hyperideals and several characterizations to classify them were provided. Many properties of ϕ prime, ϕ -primary and ϕ-δ-primary hyperideals under particular cases were investigated. In the future work, one can develop the study of ϕ-δ-primary hyperideals.



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