# On 1-Absorbing Prime Hyperideal and Some of Its Generalizations

Authors: M. Anbarloei
Journal: Journal of Mathematics
 (2022)

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## Abstract

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


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## Body

## 1. Introduction

In a classical algebraic structure, the composition of two elements is an element, but in an algebraic hyperstructure, the composition of two elements is a set. A well-known type of a hyperring is called the Krasner hyperring [1]. The hyperring is a hypercompositional structure S,+,⋅ where S,+ is a canonical hypergroup and S,⋅ is a semigroup in which the zero element is absorbing and the operation is a two-sided distributive one over the hypercomposition +. In 1982, Rota initiated the study of multiplicative hyperring [2] which was subsequently investigated by many authors [3–7]. In the hyperring, the multiplication is a hyperoperation, while the addition is an operation. The hyperring in which the additions and the multiplications are hyperoperations was introduced by Salvo in [8].In the theory of rings, the key role of the notion of prime ideal as an expansion of the notion of prime number in the ringℤ is undeniable. The notion of primeness of hyperideal in a multiplicative hyperring was conceptualized by Procesi and Rota in [6]. The notions of prime and primary hyperideals in multiplicative hyperrings were fully studied by Dasgupta in [9]. Badawi [10] introduced and studied 2-absorbing ideals and this notion is further generalized by Anderson and Badawi [11, 12]. In [13], Ghiasvand introduced the concept of 2-absorbing hyperideals in a multiplicative hyperring. Several authors have extended this concept in several ways [14–18]. The concept of 1-absorbing prime ideals which is another extension of prime ideals was introduced in [19]. The notions of 1-absorbing primary and weakly 1-absorbing primary ideals were investigated in [20].Motivated from the concept of the prime hyperideals, in this paper, we introduce and study the concept of 1-absorbing prime hyperideals and some of its generalizations in a multiplicative hyperring. Several properties of them are provided.The paper is organized as follows. In Section2, we have given some basic definitions and results of multiplicative hyperrings which we need to develop our paper. In Section 3, we introduce the concept of 1-absorbing prime hyperideals and give some basic properties of them. For example we show (Theorem 1) that if I is a 1-absorbing prime hyperideal of R, then I is a prime hyperideal of R. Furthermore, I:z is a prime hyperideal of R for every nonunit element z∈R−I. In Section 4, we study an expansion of the 1-absorbing prime hyperideals which is called 1-absorbing primary hyperideals. In particular, we discuss the relations between 1-absorbing primay hyperideals and primary hyperideals. In Section 5, the notion of strongly 1-absorbing primay hyperideals is studied. For example, it is shown (Theorem 23) that there exists a strongly 1-absorbing primary hyperideal of R if and only if 0 is a prime hyperideal or R is a local multiplicative hyperring. In the final section, we investigate the concept of weakly 1-absorbing primary hyperideals of R.


## 2. Preliminaries

Recall first the basic terms and definitions from the hyperring theory. A commutative multiplicative hyperring is an abelian groupR,+ in which a hyperoperation ∘ is defined satisfying the following:(i)
for alla,b,c∈R, we have a∘b∘c=a∘b∘c;(ii)
for alla,b,c∈R, we have a∘b+c⊆a∘b+a∘c and b+c∘a⊆b∘a+c∘a;(iii)
for alla,b∈R, we have a∘−b=−a∘b=−a∘b.(iv)
for alla,b∈R, we have a∘b=b∘a.If in (ii) the equality holds then we say that the multiplicative hyperring is strongly distributive. Throughout this paper all hyperrings are commutative multiplicative hyperrings.LetA and B be two nonempty subsets of R and r∈R. Then we define(1)A∘B=∪x∈A,y∈Bx∘y,A∘r=A∘r.A non empty subsetI of R is a hyperideal if(i)
Ifa,b∈I, then a−b∈I;(ii)
Ifx∈I and r∈R, then rox⊆I.An elemente∈R is said to be scalar identity if a=aoe for all a∈R.Definition 1.
(see [21]). A proper hyperideal I of R is maximal in R if for any hyperideal J of R with I⊊J⊆R then J=R. Also, we say that R is local, if it has just one maximal hyperideal.Definition 2.
(see [9]). A proper hyperideal P of R is called a prime hyperideal if x∘y⊆P for x,y∈R implies that x∈P or y∈P. The intersection of all prime hyperideals of R containing I is called the prime radical of I, being denoted by I. If R does not have any prime hyperideal containing I, we define I=R.
LetC=r1∘r2∘…∘rn:ri∈R,n∈ℕ. A hyperideal I of R is said to be a C-hyperideal of R if for any A∈C,A∩I≠∅ implies A⊆I. Let I be a hyperideal of R. Then, D⊆I where D=r∈R:rn⊆Ifor somen∈ℕ. The equality holds when I is a C-hyperideal of R ([9], Proposition 3.2). In this paper, we assume that all hyperideals are C-hyperideal.
Recall that a hyperidealI of R is called a strong C-hyperideal if for any E∈U, E∩I≠∅, then E⊆I, where U=∑i=1nAi:Ai∈C,n∈ℕ and C=r1∘r2∘…∘rn:ri∈R,n∈ℕ (for more details see [22]).Definition 3.
(see [9]). A non-zero proper hyperideal Q of R is called a primary hyperideal if x°y⊆Q for x,y∈R implies that x∈Q or y∈Q. Since Q=P is a prime hyperideal of R by Proposition 3.6 in [9], Q is referred to as a P-primary hyperideal of R.Definition 4.
(see [23]). Let R1,+1,∘1 and R2,+2,∘2 be two multiplicative hyperrings. A mapping from R1 into R2 is said to be a good homomorphism if for all x,y∈R1, ϕx+1y=ϕx+2ϕy and ϕx∘1y=ϕx∘2ϕy.Definition 5.
(see [21]). We call MnR as of R. Also, for all A=Aijn×n,B=Bijn×n∈P⋆MnR,A⊆B if and only if Aij⊆Bij.Definition 6.
(see [21]). An element x∈R is called unit, if there exists y∈R, such that e∈x∘y. Denote the set of all unit elements in R by UR.Definition 7.
(see [24]). A hyperring R is called an integral hyperdomain, if for all x,y∈R, 0∈x∘y implies that x=0 or y=0.Definition 8.
A hyperringR is said to be a reduced hyperring if it has no nilpotent elements. That is, if xn=0 for x∈R and a natural number n, then x=0.Example 1.
Consider the ringℤ6,⊕,⊙ that for all x¯,y¯∈ℤ6, x¯⊕y¯ and x¯⋅y¯ are the remainder of x+y/6 and x⋅y/6, respectively, which “⊕” and “⊙” are ordinary addition and multiplication, and x,y∈ℤ. The set ℤ6 with the operation ⊕ and the hyperoperation x¯⋆y¯=xy¯,2xy¯,3xy¯,4xy¯,5xy¯ for all x¯,y¯∈ℤ6 is a reduced multiplicative hyperring, since x¯n≠0 for all 0≠x¯∈ℤ6 and all n>0.Definition 9.
A multiplicative hyperringF is called a hyperfield if every non-zero element of F is unit.Example 2.
Consider the ringℤp,+,⋅ for some prime integer p. Let q∈ℤ and q≠1. We define hyperoperation ∘ on ℤp by a∘b=a⋅b,q⋅a⋅b for all a,b∈ℤp. Since every non-zero element of ℤp,+,∘ is unit, then ℤp is a hyperfield.Definition 10.
(see [25]). An element r∈R is called regular if there exists x∈R such that r∈r∘x∘r. So, R is called regular, if all of elements in R are regular elements. The set of all regular elements in R is denoted by VR.Definition 11.
LetI be a hyperideal of R and x∈R. Then define:(2)I:a=r∈R:r∘a⊆I.


## 3. 1-Absorbing Prime Hyperideals

Definition 12.
LetI be a proper hyperideal of R. I is called a 1-absorbing prime hyperideal if for nonunit elements x,y,z∈R, x∘y∘z⊆I , then either x∘y⊆I or z∈I.Example 3.
In the multiplicative hyperring of integersℤA with A=2,3 such that for any x,y∈ℤA,x∘y=x.a.y|a∈A, every principal hyperideal generated by a prime integer is a 1-absorbing prime hyperideal.
It is clear that every prime hyperideal ofR is a 1-absorbing prime hyperideal of R. Recall that a hyperideal I of R is said to be a 2-absorbing primary hyperideal if for x,y,z∈R, x∘y∘z⊆I, then xoy⊆I or x∘z⊆I or y∘z⊆I. Every 1-absorbing prime hyperideal of R is a 2-absorbing primary hyperideal of R. The following example shows that the converse may not be true, in general.Example 4.
Consider the multiplicative hyperring of integersℤA with A=3,4 such that for any x,y∈ℤA,x∘y=x.a.y|a∈A. It is easy to see 8 is a 2-absorbing primary hyperideal of ℤA. But it is not a 1-absorbing prime hyperideal of ℤA as the fact that 2∘2∘2⊆8 but 2∉8 and 2∘2=12,16⊈8.Theorem 1.
IfI is a 1-absorbing prime hyperideal of R, then I is a prime hyperideal of R. Furthermore, I:z is a prime hyperideal of R for every nonunit element z∈R−I.Proof.
Suppose thatI is a 1-absorbing prime hyperideal of R and a∘b⊆I for some nonunit a,b∈R. Therefore a∘bs⊆I for a positive integer s. Clearly, we have atas−tbs⊆I for some positive integer t<s. Since the hyperideal I of R is 1-absorbing prime, then we get either as⊆I or bs⊆I. This means either a∈I or b∈I. Consequently, the hyperideal I of R is prime. Now, let x∘y⊆I:z for some nonunit elements x,y∈R. Assume that x∉I:z. This means x∘z⊈I. Since x∘z∘y⊆I and the hyperideal I is 1-absorbing prime, then we get y∈I⊆I:z. Consequently, the hyperideal I:z is prime.
The following lemma is needed in the proof of our next result.Lemma 1.
Letu+v be a unit element of R for every nonunit element u of R and for every unit v of R. Then R is local.Proof.
LetM1 and M2 be two different maximal hyperideals of R. Hence for some m1∈M1 and m2∈M2 we have m1+m2=1. By Theorem 1 in [21], 1−m1=m2 is a unit element of R which means M2 contains a unit element. This is a contradiction. Thus R is local.Theorem 2.
If a strongC-hyperideal I of R is 1-absorbing prime that is not prime, then R is local.Proof.
LetI be a 1-absorbing prime hyperideal of R. Assume that I is not prime hyperideal. Suppose that we have x∘y⊆I for some x,y∈R but x,y∉I. Assume that a is a nonunit element and b is a unit element of R. Let a+b be a nonunit element of R. Since a∘x∘y⊆I and the hyperideal I of R is 1-absorbing prime, then we get a°x⊆I. Since a+b∘x∘y⊆I and y∉I, then we have a+b°x⊆I. Since a+b∘x⊆a∘x+b∘x and I is a strong C-hyperideal of R, we get a∘x+b∘x⊆I which implies b∘x⊆I. This means x∈I which is a contradiction. Thus a+b is a unit element of R. Hence we conclude that R is local, by Lemma 1.Theorem 3.
LetI be a 1-absorbing prime hyperideal of R. If for nonunit elements x,y∈R and all proper hyperideal J of R, x∘y∘J⊆I, then x∘y⊆I or J⊆I.Proof.
Letx∘y∘J⊆I for nonunit elements x,y∈R and all proper hyperideals J of R such that x∘y⊈I and J⊈I. Therefore there exists an element a∈J but a∉I. Then we have x∘y∘a⊆I such that x∘y⊈I and a∉I. This is a contradiction, since I is 1-absorbing hyperideal of R.Theorem 4.
LetI be a P-primary hyperideal of R such that for every c∈P−I, P2:c⊆I. Then hyperideal I of R is 1-absorbing prime.Proof.
LetI be a P-primary hyperideal of R such that for every a∈P−I, P2:a⊆I. Let for a,b,c∈R, a∘b∘c⊆I but neither a∘b⊈I nor c∉I. Therefore we have c∈P−I, because hyperideal I of R is P-primary hyperideal. Thus we get a∘b⊆P2:c⊆I, a contradiction. Then we have either a∘b⊆I or c∈I which means hyperideal I or R is 1-absorbing prime.Theorem 5.
LetI be a hyperideal of R. If MnI is a 1-absorbing hyperideal of MnR, then I is a 1-absorbing hyperideal of R.Proof.
Suppose that forx,y,z∈R , x∘y∘z⊆I. Then(3)x∘y∘z0⋯000⋯0⋮⋮⋱⋮00⋯0⊆MnI.
It is clear that(4)x∘y∘z0⋯000⋯0⋮⋮⋱⋮00⋯0=x∘y0⋯000⋯0⋮⋮⋱⋮00⋯0z0⋯000⋯0⋮⋮⋱⋮00⋯0.
SinceMnI is a 1-absorbing hyperideal of MnR, then we have(5)x∘y0⋯000⋯0⋮⋮⋱⋮00⋯0⊆MnI,z0⋯000⋯0⋮⋮⋱⋮00⋯0∈MnI.
It follows that eitherx∘y⊆I or z∈I. Therefore I is a 1-absorbing hyperideal of R.
LetR,+,o be a multiplicative hyperring. We define the relation γ as follows:
aγb if and only if a,b⊆U where U is a finite sum of finite products of elements of R, i.e.(6)aγb⟺∃z1,…,zn∈Rsuch thata,b⊆∑j∈J∏i∈Ijzi;Ij,J⊆1,…,n.
We denote the transitive closure ofγ by γ∗. The relation γ∗ is the smallest equivalence relation on a multiplicative hyperring R,+,o such that the quotient R/γ∗, the set of all equivalence classes, is a fundamental ring. Let U be the set of all finite sums of products of elements of R we can rewrite the definition of γ∗ on R as follows:(7)aγ∗b⟺∃z1,…,zn∈Rwithz1=a,zn+1=bandu1,…,un∈Usuch thatzi,zi+1⊆uifori∈1,…,n.
We suppose thatγ∗a is the equivalence class containing a∈R. Then, both the sum ⊕ and the product ⊙ in R/γ∗ are defined as follows: γ∗a⊕γ∗b=γ∗c for all c∈γ∗a+γ∗b and γ∗a⊙γ∗b=γ∗d for all d∈γ∗aoγ∗b Then R/γ∗ is a ring, which is called a fundamental ring of R [26].Theorem 6.
The hyperidealI of R is 1-absorbing hyperideal if and only if I/γ∗ is a 1-absorbing ideal of R/γ∗.Proof.
⇒ Let for x,y,z∈R/γ∗,x⊙y⊙z∈I/γ∗. Then, there exist a,b,c∈R such that x=γ∗a,y=γ∗b,z=γ∗c and x⊙y⊙z=γ∗a⊙γ∗b⊙γ∗c=γ∗a∘b∘c. Since γ∗a⊙γ∗b⊙γ∗c=γ∗a∘b∘c∈I/γ∗, then a∘b∘c⊆I. Since I is a 1-absorbing hyperideal of R, then a∘b⊆I or c∈I. Hencey x⊙y=γ∗a⊙γ∗b=γ∗a∘b∈I/γ∗ or z=γ∗c∈I/γ∗. Thus I/γ∗ is a 1-absorbing ideal of R/γ∗.
⇐ Suppose that a∘b∘c⊆I for a,b,c∈R. Then γ∗a,γ∗b,γ∗c∈R/γ∗ and γ∗a⊙γ∗b⊙γ∗c=γ∗a∘b∘c∈I/γ∗. Since I/γ∗ is a 1-absorbing ideal of R/γ∗, then we have γ∗b⊙γ∗b=γ∗a∘b∈I/γ∗ or γ∗c∈I/γ∗. It means a∘b⊆I or c∈I. Hence I is a 1-absorbing hyperideal of R. Tables 1 and 2.Table 1
H,+.
+012300123112302230133012Table 2
H,∘.
∘012300000101,321,320202301,321,3Example 5.
LetH=0,1,2,3. Consider the multiplicative hyperring H,+,∘ where “+” and “∘” are defined on H as follows:
It is easy to see thatI=0,2 is a 1-absorbing prime hyperideal of H. Then I/γ∗=γ∗0,γ∗2 is a 1-absorbing ideal of R/γ∗.Theorem 7.
LetI be a proper hyperideal of R. Then the following statements are equivalent.(1)
I is a 1-absorbing prime hyperideal of R.(2)
IfJ∘H∘K⊆I for some proper hyperideals J,H,K of R, then either J∘H⊆I or K⊆I.Proof.
1⇒2 Let I be a 1-absorbing prime hyperideal of R and J∘H∘K⊆I for some proper hyperideals J,H,K of R such that J∘H⊈I. Thus there exist nonunit elements j∈J and h∈H such that j∘h⊈I. Since j∘h∘K⊆I and j∘h⊈I, we get K⊆I by Lemma 3.7.
2⇒1 Let x∘y∘z⊆I for some nonunit elements x,y,z∈R such that x∘y⊈I. Assume that J=x, H=y and K=z. This means J∘H∘K⊆I. Since J°H⊈I, we conclude that K⊆I and so z∈I.


## 4. 1-Absorbing Primary Hyperideals

Definition 13.
LetI be a proper hyperideal of R. I is called a 1-absorbing primary hyperideal if for nonunit elements x,y,z∈R, x∘y∘z⊆I, then either xoy⊆I or z∈I.Theorem 8.
(1)
Every primary hyperideal ofR is a 1-absorbing primary hyperideal.(2)
Every 1-absorbing primary hyperideal ofR is a 2-absorbing primary hyperideal.Example 6.
In the multiplicative hyperring of integersℤA with A=2,3 such that for any x,y∈ℤA,x∘y=x.a.y|a∈A, hyperideal 5=5n|n∈ℤ is a 1-absorbing primary hyperideal.Example 7.
Letℤ,+,. be the ring of integers. We define the hyperoperation a°b=2xy,3xy for all a,b∈ℤ. Then ℤ,+,∘ is a multiplicative hyperring. The hyperideals 2 and 3 of ℤ are 1-absorbinh primary hyperideals.Theorem 9.
LetI be a hyperideal of R. If I is 1-absorbing primary, then I is a prime hyperideal of R.Proof.
Let for somex,y∈R, x∘y⊆I. We suppose that x,y are nonunit elements of R. We assume that for n≥2, x∘yn⊆I. Since x∘yn=xn∘yn=xn−1∘x∘yn⊆I, we have xn−1∘x=xn⊆I or yn⊆I, because I is a 1-absorbing primary hyperideal of R. Thus x∈I or y∈I which means I is a prime hyperideal of R.Theorem 10.
LetI be a 1-absorbing primary strong C-hyperideal of R such that is not a primary hyperideal. Then R is local.Proof.
Let hyperidealI of R be 1-absorbing primary such that is not primary. Let for some nonunit elements x,y∈R, x∘y⊆I. By the assumption, we conclude that x∉I and y∉I. Let u be a nonunit element of R. Then we get u∘x∘y⊆I. Since the hyperideal I is 1-absorbing primary and y∉I, then u∘x⊆I. Let v be a unit element of R such that u+v is a nonunit element of R. It is clear that u+v∘x∘y⊆I. Since I is a 1-absorbing primary hyperideal of R, we get u+v∘x⊆I, because y∉I. Since I is a strong C-hyperideal of R and u+v∘x⊆u∘x+v∘x, we conclude that u∘x+v∘x⊆I. Since u∘x⊆I, we have v∘x⊆I which implies x∈I which is a contradiction. Hence u+v is a unit element of R. By Lemma 1, R is local.
In the following theorem, we show that ifR is not a local multiplicative hyperring, then every 1-absorbing primary hyperideal is primary.Theorem 11.
Suppose thatR is not local and I is a strong C-hyperideal of R. Then I is a 1-absorbing primary hyperideal of R if and only if I is a primary hyperideal of R.Proof.
⇒ Let I be a 1-absorbing primary hyperideal of R and let x∘y⊆I for some nonunit elements x,y∈R such that y∉I. By Lemma 1, there exist a nonunit element u∈R and a unit element v∈R such that u+v is a nonunit element of R, because R is not local. It is clear that u∘x∘y⊆I. Since I is a 1-absorbing primary hyperideal of R and y∉I, then we get u∘x⊆I. Moreover, u+v∘x∘y⊆I. Then u+v∘x⊆I, because I is a 1-absorbing primary hyperideal of R and y∉I. Since I is a strong C-hyperideal of R and u+v∘x⊆u∘x+v∘x, we have u∘x+v∘x⊆I. Thus v∘x⊆I which implies x∈I. Consequently, I is a primary hyperideal of R.
⇐ It is clear.Theorem 12.
LetR1 and R2 be two multiplicative hyperrings. Let I be a hyperideal of R1×R2. The following statements are equivalent.(1)
I is a 1-absorbing primary hyperideal of R1×R2.(2)
I is a primary hyperideal of R1×R2.(3)
I=I1×R2 for some primary hyperideal I1 of R1.Definition 14.
A non-zero nonunit elementx of R is called irreducible if x∈x1°x2 for some x1,x2∈R, then x1 is a unit of R or x2 is a unit of R.Definition 15.
A non-zero elementx of R is called prime if x1∘x2⊆x∘r for some x1,x2,r∈R, then x1∈x∘r1 for some r1∈R or x2∈x∘r2 for some r2∈R2.Lemma 2.
LetR be a local strongly distributive multiplicative hyperring. If x is a non-zero prime element of R, then x is an irreducible element of R.Proof.
Letx is a non-zero prime element of R such that for some x1,x2∈R, x∈x1∘x2. We may assume that for some r1∈R, x1∈x∘r1. Now, we show that x2 is a unit element of R. Then we obtain x∈x∘r1∘x2. Hence 0∈x∘1−r1∘x2=x−x∘r1∘x2. Let x2 be a nonunit element of R. Since R is local, then 1−r1∘x2 is a unit element of R which implies x=0 which is a contradiction. Thus x2 is a unit element of R. Consequently, x is an irreducible element of R.Theorem 13.
LetR be a local strongly distributive multiplicative hyperring with maximal hyperideal M. If element x of M is a non-zero prime element of R with M≠x∘R, then x∘M is a 1-absorbing primary hyperideal of R that is not a primary hyperideal of R.Proof.
Let for some nonunit elementsa,b,c∈R, a∘b∘c⊆x∘M. Let a∘b⊈x∘M. Then for all r1,r2∈M, a∉x∘r1 and b∉x∘r2. Since for some r∈M, a∘b∘c⊆x∘r and for all s∈M, a∘b⊈x∘s, then we have c∈x∘t for some t∈M. Consequently, c∈x∘M=x∘R. Thus x∘M is a 1-absorbing primary hyperideal of R. Now, since M≠x∘R, then there exists m∈M but m∉x∘R. It is clear that x∘m⊆x∘M. By Lemma 2, x is an irreducible element of R, because x is a non-zero prime element of R. Then we conclude that x∉x∘M. Since x∘M=x∘R, we get m∉x∘M. Thus x∘M is not a primary hyperideal of R.Theorem 14.
LetI is a 1-absorbing primary hyperideal of R. Suppose that I is not a primary hyperideal of R. Then for some irreducible element x∈R−I and some nonunit element y∈R−I, x∘y⊆I. Moreover, if x∘y⊆I for some nonunit elements x∈R−I and y∈I, then element x of R is irreducible.Proof.
We suppose thatI is not a primary hyperideal of R. Then for some nonunit elements x∈R−I and y∈R−I, x∘y⊆I. Assume that x is not irreducible. Hence x∈a∘b for some nonunit elements a,b∈R. Since I is a 1-absorbing primary hyperideal of R and x∘y⊆a∘b∘y⊆I and y∉I, then a∘b⊆I which implies x∈I which is a contradiction. Thus x is irreducible.Theorem 15.
We suppose thatR is local with maximal hyperideal M. Let P be a prime hyperideal of R with P⊆M. Then the hyperideal P∘M of R is a 1-absorbing primary hyperideal.Proof.
Letx∘y∘z⊆P∘M for some nonunit elements x,y,z∈R. Since P∘M=M, then x∘y∘z⊆p. Let x∘y⊈P∘M. Then x∉P and y∉P which means a∘b⊈P Thus z∈P=P∘M. Consequently, hyperideal P∘M of R is a 1-absorbing primary.Theorem 16.
Let the hyperidealI be a 1-absorbing primary hyperideal of R and let a∈R−I be a nonunit element of R. Then I:a is a primary hyperideal of R.Proof.
Letx∘y⊆I:a for some nonunit elements x,y∈R. Let x∉I:a. Then a∘x⊈I. Since x∘y⊆I:a, then a∘x∘y⊆I. Since I is a 1-absorbing primary hyperideal of R and a∘x⊈I, then y∈I⊆I:a. Therefore, the hyperideal I:a is primary.Definition 16.
LetI be a 1-absorbing primary hyperideal of R. By Theorem 9, I=P is a prime hyperideal of R. Thus I is called P-1-absorbing primary hyperideal of R.Theorem 17.
LetI1,…,In be P-1-absorbing primary hyperideals of R. Then ∩i=1nIi is a P-1-absorbing primary hyperideal of R.Proof.
PutI=∩i=1nIi. It is clear that I=P. Let for some nonunit elements x,y,z∈R, x∘y∘z⊆I such that x∘y⊈I. Then there exist 1≤i≤n such that x∘y⊈Ii. Since Ii is a P-1-absorbing primary hyperideal of R, then z∈Ii=P which means z∈I. Consequently, I=∩i=1nIi is a P-1-absorbing primary hyperideal of R.Theorem 18.
LetR1 and R2 be multiplicative hyperrings and ϕ:R1⟶R2 be a good homomorphism such that if R2 is a local multiplicative hyperring, then ϕx is a nonunit of R2 for every nonunit x∈R1. Then the following statements hold:(1)
IfI2 is a 1- absorbing primary hyperideal of R2, then ϕ−1I2 is a 1- absorbing primary hyperideal of R1.(2)
Ifϕ is an epimorphism and I1 is a 1- absorbing primary hyperideal of R1 containing Kerϕ, then ϕI1 is a 1- absorbing primary hyperideal of R2.Proof.
(1) LetR2 be a local multiplicative hyperring. Let for some nonunit elements x,y,z∈R1, x∘y∘z⊆ϕ−1I2. Thus we have ϕx∘y∘z=ϕx∘ϕy∘ϕz⊆I2. Since I2 is a 1- absorbing primary hyperideal of R2, we conclude that ϕx∘y=ϕx∘ϕy⊆I2 which implies x∘y⊆ϕ−1I2 or ϕz∈I2 which implies z∈ϕ−1I2=ϕ−1I2. Therefore, ϕ−1I2 is a 1-absorbing primary hyperideal of R1. Now let I2 be a 1- absorbing primary hyperideal of R2 such that R2 is not a local multiplicative hyperring. By Theorem 10, the hyperideal I2 of R2 is primary. Then we conclude that ϕ−1I2 is a primary hyperideal of R1. Thus, ϕ−1I2 is a 1-absorbing primary hyperideal of R1.
(2) Let for some nonunit elementsx2,y2,z2∈R2, x2∘y2∘z2⊆ϕI1. Since ϕ is an epimorphism, there exist x1,y1,z1∈R1 such that ϕx1=x2,ϕy1=y2,ϕz1=z2 and so ϕx1∘y1∘z1=x2∘y2∘z2⊆ϕI1. Now take any u∈x1∘y1∘z1. Then we get ϕu∈ϕx1∘y1∘z1⊆ϕI1 and so ϕu=ϕw for some w∈I1. This implies that ϕu−w=0, that is, u−w∈Kerϕ⊆I1 and so u∈I1. Since I1 is a C-hyperideal of R1, then we conclude that x1∘y1∘z1⊆I1. Since I1 is a 1-absorbing primary hyperideal of R1 then x1∘y1⊆I1 which implies x2∘y2=ϕx1∘y1⊆ϕI1 or z1∈I1 which implies z2=ϕz1∈ϕI1=ϕI1. Consequently, ϕI1 is a 1-absorbing primary hyperideal of R2.Corollary 1.
LetI,J be two proper hyperideals of R with J⊆I such that if R/J is local, then x+J is a nonunit of for every nonunit x∈R. Then I is a 1-absorbing primary hyperideal of R if and only if I/J is a 1-absorbing primary hyperideal of R/I.Proof.
Defineϕ:R⟶R/J by ϕr=r+J. clearly, ϕ is a good epimorphism. Since Kerf=J⊆I and I is a 1-absorbing primary hyperideal of R , then the claim follows from Theorem 27 (2). Now let I/J be a 1-absorbing primary hyperideal of R/J. Then we conclude that ϕ−1I/J=I is a 1-absorbing primary hyperideal of R by Theorem 27 (1).Theorem 19.
Let the hyperidealI of R be 1-absorbing. If J is a proper hyperideal of R such that x∘y∘J⊆I for some nonunit elements x,y∈R, then x∘y⊆I or J⊆I.Proof.
Letx∘y∘J⊆I such that neither x∘y⊆I nor J⊆I. Thus there exists an element r∈J such that r∉I. It is clear that x∘y∘r⊆I but x∘y⊈I and r∉I which is a contradiction, because I is a 1-absorbing hyperideal of R.
Next, it is proved that a proper hyperidealI of R is 1-absorbing primary if and only if J∘K∘L⊆I for any proper hyperideals J,K,L of R implies that either J∘K⊆I or. L⊆ITheorem 20.
Let I be a proper hyperideal ofR. Then I is a 1-absorbing primary hyperideal of R if and only if for any proper hyperideals J,K,L of R with J∘K∘L⊆I implies that either J∘K⊆I or L⊆I.Proof.
⇒ Let the hyperideal I of R be 1-absorbing primary. Suppose that for some proper hyperideals J,K,L of R, J∘K∘L⊆I such that J∘K⊈I. This means there exist nonunit elements x∈J and y∈K but x∘y⊈I. Thus we get x∘y∘L⊆I. Since I is a 1-absorbing primary hyperideal of R, then L⊆I by Theorem 19.
⇐ Let for some nonunit elements x,y,z∈R, x∘y∘z⊆I but x∘y⊈I. We have x∘y∘z⊆I. Hence x∘y⊈I. By the assumption, we get z⊆I which means z∈I. Thus the hyperideal I of R is 1-absorbing.


## 5. Strongly 1-Absorbing Primary Hyperideals

Definition 17.
LetI be a proper hyperideal of R. I is called a strongly 1-absorbing primary hyperideal if for nonunit elements x,y,z∈R, x∘y∘z⊆I , then xoy⊆I or z∈0.
It is clear that ifI is a strongly 1-absorbing primary hyperideal of R then I is a 1-absorbing primary hyperideal. Tables 3 and 4.Table 3
Z_4,⊞.
⊞0¯1¯2¯3¯0¯0¯1¯2¯3¯1¯1¯2¯3¯0¯2¯2¯3¯0¯1¯3¯3¯0¯1¯2¯Table 4
Z_4,⊡.
⊡0¯1¯2¯3¯0¯0¯0¯0¯0¯1¯0¯Z40¯,2¯Z42¯0¯0¯,2¯0¯0¯,2¯3¯0¯Z40¯,2¯Z4Example 8.
We consider the ring0¯,1¯,2¯,3¯=ℤ4,⊞,⊙ that for each x¯,y¯∈ℤ4, x¯⊞y¯ and y¯⊙y¯ are the remainder of x+y/4 and x.y/4, respectively, which “+” and “⋅” are ordinary addition and multiplication, and x,y∈ℤ4. which ℤ4,⊞,⊡ is a multiplicative hyperring. In the hyperring, hyperideal 0,2 is a strongly 1-absorbing primary hyperideal of ℤ4.Theorem 21.
If the hyperidealsI1 and I2 of R are strongly 1-absorbing primary, then I1∩I2 is a strongly 1-absorbing primary hyperideal of R.Theorem 22.
LetI be a proper hyperideal of R. Then, I is a strongly 1-absorbing primary hyperideal of R if and only if I is 1-absorbing primary and I=0, or R is local with maximal hyperideal M=I such that M2⊆I.Proof.
⇒ It is clear that if I is a strongly 1-absorbing primary hyperideal of R, then I is a 1-absorbing primary hyperideal. Now, let I≠0. Suppose that for some x,y∈R, x∘y⊈I. We get x∘y∘z⊆I for each z∈I. Since hyperideal I is strongly 1-absorbing primary, we obtain z∈0 which implies I⊆0 which means I=0 which is a contradiction. Thus for each nonunit elements x,y∈R, x∘y⊆I. Let the hyperideal M of R be maximal. Then M2⊆I which implies M2⊆I which means M⊆I. Therefore for each maximal hyperideal M of R, M=I. Consequently, R is local.
⇐ Let I be 1-absorbing primary and I=0. Clearly, the hyperideal I is strongly 1-absorbing primary. Let R is local with the maximal hyperideal M=I such that M2⊆I. Then we have x∘y⊆M2⊆I for each nonunit elements x,y∈R. Then the hyperideal I of R is strongly 1-absorbing primary.
In view of Theorem22, we have the following results.Corollary 2.
Let the hyperidealP of R be prime. Then, P is strongly 1-absorbing primary if and only if P=0 or R is local with maximal hyperideal P.Corollary 3.
LetR is local with maximal hyperideal M and let P be a prime hyperideal of R. Then hyperideal P∘M of R is strongly 1-absorbing primary if and only if P=0 or P=M.Proof.
By Theorem8 in [27], the hyperideal P∘M of R is 1-absorbing primary. By Theorem 22, the hyperideal P∘M of R is strongly 1-absorbing primary if and only if P∘M=0 or P∘M=M and M2⊆P∘M which implies P=0 or P=M.Theorem 23.
There exists a strongly 1-absorbing primary hyperideal ofR if and only if 0 is a prime hyperideal or R is local.Proof.
⇒ Let I be a strongly 1-absorbing primary hyperideal of R. Suppose that R is not local. Then I=0, by Theorem 22. Since I is a strongly 1-absorbing primary hyperideal of R, then 0 is a prime hyperideal, by Theorem 9. ⇐ Let R is local with maximal hyperideal M. Then M is a strongly 1-absorbing primary by Corollary 2. On the other hand, if 0 is a prime hyperideal or R is local, then by Corollary 2, 0 is a strongly 1-absorbing primary of R.Corollary 4.
IfR1 and R2 are two multiplicative hyperrings. Then, R=R1×R2 has no strongly 1-absorbing primary hyperideal.Proof.
Since0R=0R1×0R2 is not a prime hyperideal in R and R is not local, then R has no strongly 1-absorbing primary hyperideal by Theorem 23.Theorem 24.
The hyperidealI of R is strongly 1-absorbing primary if and only if whenever for any proper hyperideals J,K,L of R with J∘K∘L⊆I implies that either J°K⊆I or L⊆0.Proof.
⇒ Let for some proper hyperideals J,K,L of R, J∘K∘L⊆I but J∘K⊈I. This means there exist a∈J and b∈K such that a∘b⊈I. It is clear that for each c∈L, a∘b∘c⊆I. Since I is a strongly 1-absorbing primary hyperideal of R and a∘b⊈I, we get c∈0 which implies L⊆0. ⇐ Suppose that x∘y∘z⊆I for some nonunit elements x,y,z∈R but x∘y⊈I. It is clear that x∘y∘z⊆I. Since x∘y⊈I, we have z⊆0 which implies z∈0. Thus, the hyperideal I of R is strongly 1-absorbing primary.Theorem 25.
The hyperideal0 is the only strongly 1-absorbing primary hyperideal of R if and only if R is a hyperfield or a non local integral hyperdomain.Proof.
⇒ Let R is local with maximal hyperideal M. Since the hyperideal M is strongly 1-absorbing primary, then M=0 which means R is a hyperfield. Suppose that R is not local. Since hyperideal 0 is strongly 1-absorbing primary, then we have 0=0 is a prime hyperideal which means R is an integral hyperdomain.
⇐ Clearly, if R is a hyperfield or a non local integral hyperdomain then the hyperideal 0 is the only strongly 1-absorbing primary hyperideal of R.Lemma 3.
Let the hyperidealI of R be 1-absorbing primary and let J be a proper hyperideal of R such that J⊈I. Then I:J is a primary hyperideal of R.Proof.
Let for somex,y∈R, x∘y⊆I:J but x∉I:J. Hence y is a nonunit elements of R. The element x is a nonunit element of R as well. Since x∉I:J, then for some z∈J, x∘z⊈I. Since I is a 1-absorbing primary hyperideal of R and x∘y∘z⊆I, then we have y∈I⊆I:J which means I:J is a primary hyperideal of R.Theorem 26.
Let the hyperidealI of R be 1-absorbing primary and let J be a proper hyperideal of R such that J⊈I. Then the hyperideal I:J of R is strongly 1-absorbing primary.Proof.
Let the hyperidealI of R be 1-absorbing primary and let J be a proper hyperideal of R. Suppose that I≠0. Hence R is local with maximal hyperideal I and J⊆I, which is a contradicion. Then we assume that I=0 is a prime hyperideal of R. Assume that a∈I:J. This means a∘J⊆I⊆I. Since hyperideal I of R is 1-absorbing primary and J⊈I, then a∈I. Thus, we have I⊆I:J⊆I. Consequently, I:J=I=0. Then by Lemma 3 and Theorem 22, we conclude that the hyperideal I:J of R is strongly 1-absorbing primary.
We close this section by investigation the stability of strongly 1-absorbing primary hyperideals in various ring-theoretic constructions.Theorem 27.
LetR1 and R2 be multiplicative hyperrings and ϕ:R1⟶R2 be a good homomorphism. Then the following statements hold:(1)
Ifϕ is a monomorphism and I2 is a strongly 1-absorbing primary hyperideal of R2, then ϕ−1I2 is a strongly 1-absorbing primary hyperideal of R1.(2)
Ifϕ is an epimorphism and I1 is a strongly 1-absorbing primary hyperideal of R1 containing Kerϕ, then ϕI1 is a strongly 1-absorbing primary hyperideal of R2.Proof.
(1) Let for some nonunit elementsx,y,z∈R1, x∘y∘z⊆ϕ−1I2 such that x∘y⊈ϕ−1I2. This means ϕx∘ϕy⊈I2. Since ϕx∘ϕy∘ϕz⊆I2 and the hyperideal I2 of R2 is strongly 1-absorbing primary, then ϕz∈0R2 which implies z∈0R1, because ϕ is a monomorphism. Thus ϕ−1I2 is a strongly 1-absorbing primary hyperideal of R1.
(2) Let for some nonunit elementsx2,y2,z2∈R2, x2∘y2∘z2⊆ϕI1. Since ϕ is an epimorphism, there exist x1,y1,z1∈R1 such that ϕx1=x2,ϕy1=y2,ϕz1=z2 and so ϕx1∘y1∘z1=x2∘y2∘z2⊆ϕI1. Now take any u∈x1∘y1∘z1. Then we get ϕu∈ϕx1∘y1∘z1⊆ϕI1 and so ϕu=ϕw for some w∈I1. This implies that ϕu−w=0, that is, u−w∈Kerϕ⊆I1 and so u∈I1. Since I1 is a C-hyperideal of R1, we conclude that x1∘y1∘z1⊆I1. Since I1 is a strongly 1-absorbing primary hyperideal of R1 then x1∘y1⊆I1 which implies x2∘y2=ϕx1∘y1⊆ϕI1 or z1∈0R1 which implies z2=ϕz1∈0R2 .Thus, ϕI1 is a strongly 1-absorbing primary hyperideal of R2.Corollary 5.
LetI,J be two proper hyperideals of R with J⊆I. If I is a strongly 1-absorbing primary hyperideal of R then I/J is a strongly 1-absorbing primary hyperideal of R/I.Corollary 6.
LetT be a subhyperring of R. If the hyperideal I of R is strongly 1-absorbing primary, then the hyperideal T∩I of T is strongly 1-absorbing primary.


## 6. Weakly 1-Absorbing Primary Hyperideal

Definition 18.
LetI be a proper hyperideal of R. We call I a weakly 1-absorbing primary hyperideal of R if for nonunit elements x,y,z∈R, 0∉x∘y∘z⊆I, then x∘y⊆I or z∈I.
It is clear that every 1-absorbing primary hyperideal ofR is a weakly 1-absorbing primary hyperideal. Moreover, every proper hyperideal of local multiplicative hyperring R with maximal hyperideal 0 is a weakly 1-absorbing primary hyperideal of R.Example 9.
Letℤ,+,. be the ring of integers. We define the hyperoperation a∘b=2a.b,4a.b, for all a,b∈ℤ. Then ℤ,+,∘ is a multiplicative hyperring. The subset 15=15n|n∈ℤ is a weakly 1-absorbing primary hyperideal of ℤ.Theorem 28.
LetI be a weakly 1-absorbing primary ideal of R. If the hyperideal I of R is maximal, then I is primary and then I is 1-absorbing primary.Proof.
Assume thatI is a weakly 1-absorbing primary hyperideal of R such that the hyperideal I of R is maximal. This means that I is a prime hyperideal of R. Hence I is a primary hyperideal of R. Consequently, the hyperideal I of R is 1-absorbing primary.Theorem 29.
LetR be a reduced hyperring and let I be non-zero hyperideal of R. If I is weakly 1-absorbing primary, then I is a prime hyperideal of R.Proof.
Assume that for some nonunit elementsx,y∈R, 0≠x∘y⊆I. This means that x∘yn⊆I for n=2tt≥1. Since R is a reduced hyperring x∘yn≠0. we have 0≠xt∘xt∘yn⊆I. Since I is a weakly 1-absorbing primary hyperideal of R, then we get xt∘xt=xn⊆I or yn⊆I. Since I≠0, then I is a prime hyperideal of R.Theorem 30.
LetI be a non-zero hyperideal of a regular hyperring R. Then the followings are equivalent:(1)
The hyperidealI of R is weakly 1-absorbing primary.(2)
The hyperidealI of R is primary.(3)
The hyperidealI of R is 1-absorbing primary.Proof.
1⇒2 Since R is a regular hyperring, then R is a reduced hyperring, by Theorem 8 in [25]. Therefore, by Theorem 29, the hyperideal I of R is prime. Since every prime hyperideal in a regular hyperring is maximal, then hyperideal I of R is maximal. Thus the hyperideal I of R is primary by Theorem 28. The proof of other cases is clear.Definition 19.
Let the hyperidealI of R be weakly 1-absorbing primary and x,y,z be nonunit elements of R. We call x,y,z is a 1-triple-zero of I if 0=x∘y∘z , x∘y⊈I and z∉I.Theorem 31.
LetR be a strongly distributive multiplicative hyperring. Let the hyperideal I of R be weakly 1-absorbing primary and x,y,z be a 1-triple-zero of I. Then(1)
0∈x∘y∘I.(2)
Ifx,y∉I:z, then 0∈y∘z∘I and 0∈x∘z∘I.Proof.
(1) Let0∉x∘y∘I. Then for every nonunit element u∈I, 0∉x∘y∘u. Therefore we have 0∉x∘y∘z+u⊆I. Since x,y,z is a 1-triple-zero and I is a weakly 1-absorbing primary hyperideal of R, then z+u∈I. Since u∈I⊆I, then z∈I which is a contradiction. Hence, 0∈x∘y∘I.
(2) Let0∉y∘z∘I. Then for every nonunit element v∈I such that 0∉y∘z∘v. Therefore we have 0∉x+v∘y∘z⊆I. Since y∘z⊈I, then x+v is a nonunit element of R. Since x∘y⊈I and v∘y⊆I and the hyperideal I of R is weakly 1-absorbing primary, then x+v∘y⊈I which implies z∈I which is a contradiction. By a similar argument, we can conclude that 0∈x∘z∘I.Theorem 32.
LetIii∈Δ be a set of weakly 1-absorbing primary hyperideals of R such that for every distinct i,j∈Δ, P=Ii=Ij. Then ∩i∈ΔIi is a weakly 1-absorbing primary hyperideal of R.Proof.
We assume thatI=∩i∈ΔIi. Let for nonunit elements x,y,z∈R, 0≠x∘y∘z⊆I such that x∘y⊈I. This means that there exists t∈Δ, 0≠x∘y∘z⊆It but x∘y⊈It. Since It is a weakly 1-absorbing primary hyperideal of R, then z∈It=P=I.Theorem 33.
LetR1 and R2 be multiplicative hyperrings with identity such that are not hyperfields. Let I be a non-zero proper hyperideal of R1×R2. Then the followings are equivalent.(1)
I is a weakly 1-absorbing primary hyperideal of R1×R2.(2)
I=R1×I2 for some primary hyperideal I2 of R2 or I=I1×R2 for some primary hyperideal I1 of R1.(3)
I is a 1-absorbing primary hyperideal of R1×R2.(4)
I is a primary hyperideal of R1.Proof.
1⇒2 Let the hyperideal I of R1×R2 is weakly 1-absorbing primary. Suppose that for some hyperideals I1 of R1 and I2 of R2, we have I=I1×I2. Take any 0≠z∈I1. We get 0≠z,0⊆eR1,0∘eR1,0∘z,eR2 which implies eR1,0∘eR1,0∘z,eR2⊆I1×I2. Then eR1,0∘eR1,0⊆I1×I2 which implies eR1∈I1 or z,eR2∈I1×I2=I1×I2 which implies eR2∈I2. This is a contradiction. Hence, we assume that for some proper hyperideal I=I1×R2. Let for some nonunit x1,y1∈R1, x1⋆y1⊆I1. Let x2 be a nonunit element of R2. Since 0≠x1⋆y1,x2⊆x1,eR2∘eR1,x2∘y1,eR2, then x1,eR2∘eR1,x2∘y1,eR2⊆I1×R2. This means x1,eR2∘eR1,x2⊆I1×R2 which implies x1∈I1 or y1,eR2∈I1×R2=I1×R2 which implies y1∈I1.2⇒3 The proof follows from Theorem 8 (1).3⇒4 The proof follows from Theorem 11.4⇒1 It is obvious.Theorem 34.
LetR1 and R2 be multiplicative hyperrings and ϕ:R1⟶R2 be a good homomorphism. Then the followings hold:(1)
Ifϕ is a monomorphism and for every nonunit element x∈R1, ϕx is a nonunit element of R2 and I2 is a weakly 1-absorbing primary hyperideal of R2, then ϕ−1I2 is a weakly 1-absorbing primary hyperideal of R1.(2)
Ifϕ is an epimorphism and I1 is a weakly 1-absorbing primary hyperideal of R1 containing Kerϕ, then ϕI1 is a weakly 1-absorbing primary hyperideal of R2.Proof.
(1) Let for some nonunit elementsx,y,z∈R1, 0≠x∘y∘z⊆ϕ−1I2. Since ϕ is a monomorphism, then 0≠ϕx∘ϕy∘ϕz⊆I2. Since hyperideal I2 of R2 is weakly 1-absorbing primary, then ϕx∘ϕy⊆I2 which implies x∘y⊆ϕ−1I2 or ϕz∈I2 which implies z∈ϕ−1I2=ϕ−1I2. Thus ϕ−1I2 is a weakly 1-absorbing primary hyperideal of R1.
(2) Let for some nonunit elementsx2,y2,z2∈R2, 0≠x2∘y2∘z2⊆ϕI1. Since ϕ is an epimorphism, there exist x1,y1,z1∈R1 such that ϕx1=x2,ϕy1=y2,ϕz1=z2 and so ϕx1∘y1∘z1=x2∘y2∘z2⊆ϕI1. Now take any u∈x1∘y1∘z1. Then we get ϕu∈ϕx1∘y1∘z1⊆ϕI1 and so ϕu=ϕw for some w∈I1. This implies that ϕu−w=0, that is, u−w∈Kerϕ⊆I1 and so u∈I1. Since I1 is a C-hyperideal of R1, then we conclude that 0≠x1∘y1∘z1⊆I1. Since I1 is a weakly 1-absorbing primary hyperideal of R1, then x1∘y1⊆I1 which implies x2∘y2=ϕx1∘y1⊆ϕI1 or z1∈I1 which implies z2=ϕz1∈ϕI1=ϕI1 .Thus, ϕI1 is a weakly 1-absorbing primary hyperideal of R2.Corollary 7.
LetI and J be proper hyperideals of R such that J⊆I. If I is a weakly 1-absorbing primary hyperideal of R, then I/J is a weakly 1-absorbing primary ideal of R/J.Proof.
Defineϕ:R⟶R/I by ϕr=r+J. It is clear that ϕ is a good epimorphism. Since Kerϕ=J⊆I and I is a weakly 1-absorbing primary hyperideal of R, then the claim follows from Theorem 34 (2).Definition 20.
Let the hyperidealI of R be weakly 1-absorbing primary and for some proper hyperideals J,H,K of R, J∘H∘K⊆I. If for every j∈J, h∈H, k∈K, j,h,k is not a 1-triple zero, then I is called a free 1-triple zero with respect to J∘H∘K.Lemma 4.
Let the hyperidealI of R be weakly 1-absorbing primary and K be a proper hyperideal of R such that for some x,y∈R, x∘y∘K⊆I. If for all k∈K, x,y,k is not a 1-triple zero of I and x∘y⊈I, then K⊆I.Proof.
LetK⊆I. This means that there exists k∈K but k∉I. Then we have x∘y∘k⊆x∘y∘K⊆I. Since x,y,k is not a 1-triple zero and x∘y⊈I and x∘y∘z=0, then we get k∈I which is a contradiction. Hence, K⊆I.Theorem 35.
Let the hyperidealI of R be weakly 1-absorbing primary and for some proper hyperideals J,H,K of R, 0≠J∘H∘K⊆I. If I is free 1-triple zero with respect to J∘H∘K, then J∘H⊆I or K⊆I.Proof.
Let for proper hyperidealsJ,H,K of R, 0≠J∘H∘K⊆I such that J∘H⊈I. This means that there exist j∈J and h∈H such that j∘h⊈I. For all k∈K, j,h,k is not a 1-triple zero of I, because I is a free 1-triple zero with respect to J∘H∘K. By Lemma 1.13, we conclude that K⊆I.


## 7. Conclusion

In this study, we defined a generalization of prime hyperideals which is called 1-absorbing prime hyperideal and extended this concept to 1-absorbing primary hyperideals, strongly 1-absorbing primary hyperideals and weakly 1-absorbing primary hyperideals over the commutative multiplicative hyperrings. We explained the overall framework of these structures and investigated some of their essential characteristics. Many results were given to show the relations between these new concepts and other hyperideals.



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