# 2-Prime Hyperideals of Multiplicative Hyperrings

Authors: Mahdi Anbarloei
Journal: Journal of Mathematics
 (2022)

---

## Abstract

Multiplicative hyperrings are an important class of algebraic hyperstructures which generalize rings further to allow multiple output values for the multiplication operation. LetR be a commutative multiplicative hyperring. A proper hyperideal I of R is called 2-prime if x∘y⊆I for some x,y∈R, then, x2⊆I or y2⊆I. The 2-prime hyperideals are a generalization of prime hyperideals. In this paper, we aim to study 2-prime hyperideals and give some results. Moreover, we investigate δ-2-primary hyperideals which are an expansion of 2-prime hyperideals.


---

## Body

## 1. Introduction

The theory of algebraic hyperstructures was first introduced by Marty [1]. He defined the hypergroups as a generalization of groups in 1934. Since then, algebraic hyperstructures have been investigated by many researchers with numerous applications in both pure and applied sciences [2–9]. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Similar to hypergroups, hyperrings are algebraic structures more general than rings, substituting both or only one of the binary operations of addition and multiplication by hyperoperations. The hyperrings were introduced and studied by many authors [10–13]. Krasner introduced a type of the hyperring where addition is a hyperoperation and multiplication is an ordinary binary operation. Such a hyperring is called a Krasner hyperring [14]. A well-known type of a hyperring, called the multiplicative hyperring. The hyperring was introduced by Rota in 1982 which the multiplication is a hyperoperation, while the addition is an operation [15]. There exists a general type of hyperrings that both the addition and multiplication are hyperoperations [16]. Ameri and Kordi have studied Von Neumann regularity in multiplicative hyperrings [17]. Moreover, they introduced the concept of clean multiplicative hyperrings and studied some topological concepts to realize clean elements of a multiplicative hyperring by clopen subsets of its Zariski topology [18]. The notions such as (weak) zero divisor, (weak) nilpotent and unit in an arbitrary commutative hyperrings were introduced in the study by Ameri and Norouzi [19]. Some equivalence relations—called fundamental relations—play important roles in the theory of algebraic hyperstructures. The fundamental relations are one of the most important and interesting concepts in algebraic hyperstructures that ordinary algebraic structures are derived from algebraic hyperstructures by them. For more details about hyperrings and fundamental relations on hyperrings, see [1, 16, 20–23]. Prime ideals and primary ideals are two of the most important structures in commutative algebra. The notion of primeness of hyperideal in a multiplicative hyperring was conceptualized by Procesi and Rota [24]. Dasgupta extended the prime and primary hyperideals in multiplicative hyperrings [25]. Beddani and Messirdi [26] introduced a generalization of prime ideals called 2-prime ideals, and this idea is further generalized by Ulucak and Çelikel [27]. In [28], Dongsheng defined a new notion which is called δ-primary ideals in commutative rings. In [29], Ulucak introduced the concepts δ-primary and 2-absorbing δ-primary hyperideal over multiplicative hyperrings. In [30], we investigate δ-primary hyperideals in a Krasner m,n-hyperring which unify prime hyperideals and primary hyperideals.In this paper, we consider the class of multiplicative hyperring as a hyperstructureR,+,∘, where R,+ is an abelian group, R,∘ is a semihypergroup, and the hyperoperation “∘” is distributive with respect to the operation “+.” In this paper, we introduce and study the notion of 2-prime hyperideals of multiplicative hyperrings which are a generalization of prime hyperideals. Several properties of them are provided. Moreover, we investigate δ-2-primary hyperideals which are an expansion of 2-prime hyperideals. An earlier version of the manuscript has been presented as a preprint in https://www.researchgate.net [31].


## 2. Preliminaries

In this section, we give some definitions and results which we need to develop our paper.A hyperoperation “∘” on a nonempty set G is a mapping of G×G into the family of all nonempty subsets of G. Assume that “∘” is a hyperoperation on G. Then, G,∘ is called hypergroupoid. The hyperoperation on G can be extended to subsets of G as follows. Let X,Y be subsets of G and g∈G, then,(1)X∘Y=∪x∈X,y∈Yx∘y,X∘g=X∘g.A hypergroupoidG,∘ is called a semihypergroup if for all x,y,z∈G, x∘y∘z=x∘y∘z, which is associative. A semihypergroup is said to be a hypergroup if g∘G=G=G∘g for all g∈G. A nonempty subset H of a semihypergroup G,∘ is called a subhypergroup if for all x∈H we have x∘H=H=H∘x. A commutative hypergroup G,∘ is canonical if(i)
there existse∈G with e∘x=x, for every x∈G;(ii)
for everyx∈G there exists a unique x−1∈G with e∈x∘x−1; and(iii)
x∈y∘z implies y∈x∘z−1.A nonempty setR with two hyperoperations “+” and “∘” is called a hyperring if R,+ is a canonical hypergroup, R,∘ is a semihypergroup with r∘0=0∘r=0 for all r∈R and the hyperoperation “∘” is distributive with respect to +, i.e., x∘y+z=x∘y+x∘z and x+y∘z=x∘z+y∘z for all x,y,z∈R.Definition 1.
A 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; and(iii)
for alla,b∈R, we have a∘−b=−a∘b=−a∘b.
If in (ii) the equality holds, then, we say that the multiplicative hyperring is strongly distributive. A nonempty subsetI of a multiplicative hyperring R is a hyperideal,(i)
Ifa,b∈I, then a−b∈I;(ii)
Ifx∈I and r∈R, then r∘x⊆I.
Letℤ,+,⋅ be the ring of integers. Corresponding to every subset A∈P⋆ℤ such that A≥2, there exists a multiplicative hyperring ℤA,+,∘ with ℤA=ℤ and for any a,b∈ℤA, a∘b=a⋅r⋅b|r∈A.Definition 2 (see [25]).
A proper hyperidealP of a multiplicative hyperring 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 the multiplicative hyperring R does not have any prime hyperideal containing I, we define I=R.Definition 3 (see [32]).
A proper hyperidealI of a multiplicative hyperring 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 a local multiplicative hyperring, if it has just one maximal hyperideal.
LetC be the class of all finite products of elements of R, i.e., C=r1∘r2∘⋯∘rn:ri∈R,n∈ℕ⊆P∗R. 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 ([25], Proposition 1). In this paper, we assume that all hyperideals are C-hyperideal.Definition 4 (see [25]).
A nonzero proper hyperidealQ of a multiplicative hyperring 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 [25], Q is referred to as a P-primary hyperideal of R.Definition 5 (see [32]).
LetR be a commutative multiplicative hyperring and e be an identity (i. e., for all a∈R, a∈a∘e). An element x∈R is called unit, if there exists y∈R, such that e∈x∘y.Definition 6.
A hyperringR is called an integral hyperdomain, if for all x,y∈R, 0∈x⋅y implies that x=0 or y=0.Definition 7 (see [32]).
LetR be a multiplicative hyperring. The element x∈R is an idempotent if x∈x2.Definition 8 (see [19]).
An elementa∈R is said to be zero divisor if there exists 0≠b∈R such that 0=a∘b.Definition 9.
LetR1,+1,∘1 and R2,+2,∘2 be two multiplicative hyperrings. A mapping f from R1 into R2 is said to be a good homomorphism if for all x,y∈R1, fx+1y=fx+2fy, and fx∘1y=fx∘2fy.Definition 10 (see [33]).
A functionδ is called a hyperideal expansion of R if it assigns to each hyperideal I of R a hyperideal δI such that it has the following properties:(1)
I⊆δI.(2)
IfI⊆J for any hyperideals I,J of R, then, δI⊆δJ.
For example, consider the hyperideal expansionsδ0, δ1, δ+, and δ∗ of R defined with δ0I=I, δ1I=I, δ⋆I=I+J (for some hyperideal J of R), and δ∗I=I:K (for some hyperideal K of R) for all hyperideals I of R, respectively. Also, let δ be a hyperideal expansion of R and I,J two hyperideals of R such that I⊆J. Let δq:R/I⟶R/I be defined by δqJ/I=δJ/I. Then, δq is a hyperideal expansion of R/I.Definition 11.
(See [29]) Let δ be a hyperideal expansion of R. A hyperideal I of R is called a δ-primary hyperideal if x,y∈R and x∘y⊆I imply either x∈I or y∈δI.Definition 12.
(See [29]) Let f:R1⟶R2 be a good hyperring homomorphism, δ and γ hyperideal expansions of R1 and R2, respectively. Then, f is called a δγ-homomorphism if δf−1I2=f−1γI2 for each hyperideal I2 of R2.
Moreover, iff is a δγ-epimorphism and I is a hyperideal of R with Kerf⊆I, then, γfI=fδI.


## 3. 2-Prime Hyperideals

Definition 13.
LetI be a proper hyperideal of a multiplicative hyperring R. We say that I is 2-prime if for all x,y∈R, x∘y⊆I implies x2⊆I or y2⊆I.Example 1.
Letℤ,+,⋅ be the ring of integers. In the multiplicative hyperring ℤA,+,∘ with A=5,7 and the hyperoperation a∘b=a⋅r⋅b|r∈A for a,b∈ℤA, the principal hyperideal 3ℤ is 2-prime.Example 2.
Consider the ring0¯,1¯,2¯,3¯=ℤ,⊞,⊙ 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∈ℤ (see Tables 1 and 2)
ℤ4,⊞,⊡ is a multiplicative hyperring. In the hyperring, hyperideal 0,2 is a 2-prime hyperideal of ℤ4.
Note that an intersection of the 2-prime hyperideals may not be a 2-prime hyperideal ofR.Table 1
Addition operation.
⊞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 2
Multiplication hyperoperation.
⊡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 3.
Consider the ring of integersℤ. We define a∘b=2ab,3ab for all a,b∈ℤ. Then, ℤ,+,∘ is a multiplicative hyperring. The hyperideals 5ℤ and 7ℤ are 2-prime, but 5ℤ∩7ℤ=35ℤ since 5∘7⊆35ℤ and 52,72⊈35ℤ.Theorem 1.
LetI be a hyperideal of a multiplicative hyperring R. Then,(1)
IfI is a 2-prime hyperideal, then I is a prime hyperideal.(2)
IfP is a prime hyperideal of R, then P2 is a 2-prime hyperideal of R.(3)
LetR1 and R2 be two multiplicative hyperrings such that f:R1⟶R2 is a good homomorphism. If I2 is a 2-prime hyperideal of R2, then f−1I2 is a 2-prime hyperideals of R1.(4)
LetR1 and R2 be two multiplicative hyperrings such that f:R1⟶R2 is a good epimorphism. If I1 is a 2-prime hyperideal of R1 with Kerf⊆I1, then, fI1 is a 2-prime hyperideal of R2.(5)
IfI is a 2-prime hyperideal and I1,I2 are two subsets of R with I1°I2⊆I, then, ∪x∈I1x2⊆I or ∪y∈I2y2⊆I.Proof.
(1)
Let the hyperidealI be 2-prime. If x∘y⊆I for x,y∈R, then, for some positive integer n, xn∘yn⊆I. Since hyperideal I is 2-prime, we have x2n⊆I or y2n⊆I. Thus, we get x∈I or y∈I which means hyperideal I is prime.(2)
SinceP2⊆P, then, we are done.(3)
Letx∘y⊆f−1I2 for x,y∈R1. Hence, fx∘y⊆I2. Since the hyperideal I2 is 2-prime, then, f2x=fx2⊆I2 or f2y=fy2⊆I2. Thus, x2⊆f−1I2 or y2⊆f−1I2, i.e., the hyperideal f−1I2 is 2-prime.(4)
Let for some elementsx2,y2∈R2, x2∘y2⊆fI1. Since f is an epimorphism, there exist x1,y1∈R1 such that fx1=x2,fy1=y2 and so fx1∘y1=x2∘y2⊆fI1. Now take any u∈x1∘y1. Then, we get fu∈fx1∘y1⊆fI1 and so fu=fw for some w∈I1. This implies that fu−w=0, that is, u−w∈Kerf⊆I1 and so u∈I1. Since I1 is a C-hyperideal of R1, then, we conclude that x1∘y1⊆I1. Since I1 is a 2-prime hyperideal of R1 then, x12⊆I1 which implies x22=fx12⊆fI1 or y12⊆I1 which implies y22=fy12⊆fI1. Consequently, fI1 is a 2-prime hyperideal of R2.(5)
Let∪x∈I1x2⊈I or ∪y∈I2y2⊈I. Hence, there exist x0∈I1 and y0∈I2 with x02⊈I and y02⊈I. Since hyperideal I is 2-prime, then, x0∘y0⊈I which is a contradiction.
LetI be a 2-prime hyperideal of R. Since I=P is a prime hyperideal of R by Theorem 1. (1), I is referred to as a P-2-prime hyperideal of R.Theorem 2.
LetI be a P-2-prime hyperideal of a multiplicative hyperring R. Then, I:x2 for all x∈R−I is a P-2-prime hyperideal of R.Proof.
Lety∈I:x2 for x∈R−P. This means that y∘x2⊆I. Since the hyperideal I is P-2-prime and x∉P=I, then, y2⊆I which means y∈P. Therefore, we have I:x2⊆P. Since I⊆I:x2⊆P, then, I⊆I:x2⊆P. Since I=P=P, then, we have I:x2=P. Assume that for u,w∈R, u∘w⊆I:x2 such that w2⊈I:x2. We have u∘w∘x2=u∘x∘w∘x⊆I. Since the hyperideal I is P-2-prime and w∘x2⊈I, then, u∘x2=u2∘x2⊆I. This means u2⊆I:x2. Consequently, I:x2 is a P-2-prime hyperideal of R.
In reference [17], recall that an element x∈R is called regular if there exists r∈R such that x∈x2∘r. So, we can define that R is regular multiplicative hyperring, if all of elements in R are regular elements.Theorem 3.
LetR be a regular multiplicative hyperring and I be a 2-prime hyperideal of R. Then, I is prime.Proof.
LetI be a 2-prime hyperideal of R. Suppose that a∘b⊆I with a∉I for some a,b∈R. Since R is regular, then, there exists r∈R such that a∈a2∘r. Let a2⊆I. Then, a∈a2∘r⊆I which is a contradiction. Therefore, a2⊈I. Since I is a 2-prime hyperideal of R, then, b2⊆I. This means b∈I=I. Consequently, the hyperideal I is prime.
Recall that a proper hyperidealI of a multiplicative hyperring R is called semiprime if whenever xk∘y⊆I for some x,y∈R and k∈ℤ+, then, x∘y⊆I. Note that if I is a semiprime hyperideal of R such that Jn⊆I for some hyperideal J of R and n∈ℕ, then, J⊆I (Proposition 2 in [34]). Every prime hyperideal is a semiprime hyperideal, but, the converse is not true in general. For example, the hyperideal 6 of ℤ30 is semiprime, but it is not prime (see Example 2.3 in [34]).Theorem 4.
LetI be a semiprime hyperideal of a multiplicative hyperring R and δ be a hyperideal expansion of R. Then, I is a 2-prime hyperideal of R if and only if I is a prime hyperideal of R.Definition 14.
A proper hyperidealI of a multiplicative hyperring R is called semiprimary if for all x,y∈R such that x∘y⊆I, then, either x or y lies in I.Example 4.
Letℤ,+,⋅ be the ring of integers. In the multiplicative hyperring ℤA,+,∘ with A=2,3 and hyperoperation a∘b=a⋅r⋅b|r∈A for a,b∈ℤA, the hyperideal 11ℤ is semiprimary.Theorem 5.
Every 2-prime hyperideal ofR is a semiprimary hyperideal of R.Theorem 6.
LetℤA be a multiplicative hyperring of integers and a>1 be a positive integer such that each element x of A and a are coprime and x2∈A. Then, I=a is a 2-prime hyperideal of ℤA if and only if I=pn for some positive integer n and an irreducible element p of ℤA or p=0.Proof.
⇒ Let I=a be a 2-prime hyperideal of ℤA. Assume that a is not an irreducible element. Then, a=p1m1p2m2,…,pnmn is a representation of a as a product of distinct prime integers pi such that mi is a positive integer for 1≤i≤n. Put x=p1m1 and y=p2m2,…,pnmn by Proposition 4.13 in [25]. For every c∈A, xcy∈a which implies x∘y⊆a. Since a is 2-prime, then, we have x2⊆a or y2⊆a. If x2⊆a, then, for all d∈A we have x2d∈a. Thus, we obtain x2d=p12m1d=αa=αp1m1,p2m2,…,pnmn for some α∈ℤA. Since each element of A and a is coprime, then, for some 2≤i≤n, we get pi|p1 which is contradiction. If y2⊆a then for every e∈A, y2e∈a which implies y2e=p2m2,…,pnmn2e=βa=p1m1,p2m2,…,pnmn for β∈ℤ which means p1|pj for some 2≤j≤n which is a contradiction.
⇐ Let I=pn for some positive integer n and an irreducible element p∈ℤA. Let x∘y⊆I for some x,y∈ℤ. Then, for α∈A, xyα∈I which implies x=aps and yα=bpt for some a,b∈ℤ with n≤t+s. If 2s<n and 2t<n, then, 2s+2t<2n which is a contradiction. Therefore, x2β∈I for some β∈A or y2α2∈I which implies x2⊆I or y2⊆I. Consequently, I is a 2-prime hyperideal of ℤ.
LetR be a multiplicative hyperring. Then, we call MnR as the set of all hypermatixes of R. Also, for all A=Aijn×n,B=Bijn×n∈P⋆MnR,A⊆B if and only if Aij⊆Bij [32].Theorem 7.
LetR be a multiplicative hyperring with scalar identity 1 and I be a hyperideal of R. If MnI is a 2-prime hyperideal of MnR, then, I is an 2-prime hyperideal of R.Proof.
Suppose that forx,y∈R , x∘y⊆I. Then,(2)x∘y0⋯000⋯0⋮⋮⋱⋮00⋯0⊆MnI,
It is clear that(3)x∘y0⋯000⋯0⋮⋮⋱⋮00⋯0=x0⋯000⋯0⋮⋮⋱⋮00⋯0y0⋯000⋯0⋮⋮⋱⋮00⋯0,since MnI is a 2-prime hyperideal of MnR, then,(4)x0⋯000⋯0⋮⋮⋱⋮00⋯02=x20⋯000⋯0⋮⋮⋱⋮00⋯0⊆MnI,which means x2⊆I or(5)y0⋯000⋯0⋮⋮⋱⋮00⋯02=y20⋯000⋯0⋮⋮⋱⋮00⋯0⊆MnI,which means y2⊆I. Therefore, I is a 2-prime hyperideal of R.
LetR,+,∘ be a 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., aγb⇔∃z1,…,zn∈R such that a,b⊆∑j∈J∏i∈Ijzi;Ij,J⊆1,…,ndy. We denote the transitive closure of γ by γ∗. The relation γ∗ is the smallest equivalence relation on a multiplicative hyperring R,+,∘ 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:
aγ∗b if and only if there exist z1,…,zn∈R with z1=a,zn+1=b and u1,…,un∈U such that zi,zi+1⊆ui for i∈1,…,n. Suppose that γ∗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∗a∘γ∗b Then, R/γ∗ is a ring, which is called a fundamental ring of R (see also [9]).Theorem 8.
LetR be a multiplicative hyperring with scalar identity 1. Then, a hyperideal I of R is 2-prime if and only if I/γ∗ is a 2-prime ideal of R/γ∗.Proof.
⇒ Let for x,y∈R/γ∗,x⊙y∈I/γ∗. Thus, there exist a,b∈R such that x=γ∗a,y=γ∗b and x⊙y=γ∗a⊙γ∗b=γ∗a°b. So, γ∗a⊙γ∗b=γ∗a∘b∈I/γ∗, then, a∘b⊆I. Since I is 2-prime hyperideal, then, a2⊆I or b2⊆I. Hence, x2=γ∗a2∈I/γ∗ or y2=γ∗b2∈I/γ∗. Thus, I/γ∗ is a 2-prime ideal of R/γ∗. ⇐ Suppose that a∘b⊆I for a,b∈R, then, γ∗a,γ∗b∈R/γ∗ and γ∗a⊙γ∗b=γ∗a∘b∈I/γ∗. Since I/γ∗ is a 2-prime ideal of R/γ∗, then, we have γ∗a2∈I/γ∗ or γ∗b2∈I/γ∗. It means that a2⊆I or b2⊆I. Hence, I is a 2-prime hyperideal of R.Lemma 1.
LetR be a local multiplicative hyperring with maximal hyperideal M and P be a prime hyperideal of R. Then, P∘M is a 2-prime hyperideal of R. Moreover, if P∘M is a prime hyperideal of R, then, P∘M=P.Proof.
Letx∘y⊆P∘M⊆P for some x,y∈R. This means x∈P or y∈P. Suppose that x∈P. Clearly, x is not unit. Then, we have x∈M which implies x2⊆P∘M. Thus, the hyperideal P∘M is 2-prime.
For the second assertion, assume that the hyperidealP∘M is prime. Suppose that x∈P. Since P⊆M, then, x2⊆P∘M which implies x∈P∘M because P∘M is a prime hyperideal of R. This means P⊆P∘M. Since P∘M⊆P, then, we have P∘M=P.Theorem 9.
LetR be a local multiplicative hyperring with maximal hyperideal M. Then, every 2-prime hyperideal of R is prime if and only if for each minimal prime hyperideal P over an arbitrary 2-prime hyperideal I, I∘M=P. Furthermore, if every 2-prime hyperideal of R is prime, then, M is an idempotent hyperideal.Proof.
⇒ Assume that every 2-prime hyperideal of R is prime. Since the hyperideal I is 2-prime, then, I=P is prime. Then, I∘M is a 2-prime hyperideal of R, by Lemma 1. This means I∘M is a prime hyperideal of R. Then, we conclude that I∘M=I, by Lemma 1.
⇐ Suppose that I is a 2-prime hyperideal of R. Then, we have I⊆I=P, by Theorem 1 (1). By the assumption, we get P=I∘M⊆I∩M=I. This implies that I=P is a prime hyperideal of R.Theorem 10.
Let every 2-prime hyperideal of a multiplicative hyperringR be prime and P be an arbitrary prime hyperideal of R. Then, P2=P.Proof.
Assume that every 2-prime hyperideal ofR is prime and P is an arbitrary prime hyperideal of R. Then, the hyperideal P2 is 2-prime, by Theorem 1 (2). Since very 2-prime hyperideal of R is prime, then, the hyperideal P2 is prime. It is easy to see that P2 is equal to P.Definition 15.
LetI be a hyperideal of a multiplicative hyperring R. We say that a 2-prime hyperideal P is minimal over I if there is no a 2-prime hyperideal P′ of R with I⊆P′⊂P. Note that 2-MinRI denotes the set of minimal 2-prime hyperideals over I.Theorem 11.
LetR be a local multiplicative hyperring with maximal hyperideal M and P be a prime hyperideal of R. Let for each 2-prime hyperideal I of R, I2⊆I. Then, the followings are equivalent:(1)
IfP∈MinRI for each hyperideal I∈2−MinRP2, then, I∘M=P.(2)
IfI⊆P for each hyperideal I∈2−MinRP2, then, I=P.Proof.
(1)⇒ (2) Suppose that I⊆P for some I∈2−MinRP2. First, we show that the hyperideal P is minimal over I. Assume that we have I⊆P′⊆P for some prime hyperideal P′ of R. By the assumption, we get P2⊆I⊆P′⊆P. Let a∈P. Then, we obtain a2⊆P2 which implies a2⊆P′. This means a∈P′, since P′ is a prime hyperideal of R. Thus, we conclude that the hyperideal P is minimal over I. Now, since I∘M⊆I⊆P and I∘M=P, then, we get I=P.
(2)⇒ (1) Let P∈MinRI such that I∈2−MinRP2. We get I=P since the hyperideal I is prime. Clearly, I2=P2⊆I⊆P. By Lemma 1, P∘M is a 2-prime hyperideal of R. Since P2⊆P∘M⊆I and I=P, then, P∘M=I∘M=P.Corollary 1.
LetR be a local multiplicative hyperring with maximal hyperideal M and P be a prime hyperideal of R. Let for each P-2-prime hyperideal I, I∈2−MinRP2. Then, for each hyperideal I∈2−MinRP2 with I⊆P, I=P if and only if every 2-prime hyperideal of R is prime.Proof.
The claim follows by Theorems9 and 11.


## 4. Expansion of 2-Prime Hyperideals

Definition 16.
Letδ be a hyperideal expansion of a multiplicative hyperring R. A proper hyperideal I of R is called δ-2-primary if for a,b∈R, a∘b⊆I implies either a2⊆I or b2⊆δI.Example 5.
Suppose thatℤ,+,⋅ is the ring of integers. For all a,b∈ℤ, we define the hyperoperation a∘b=a⋅x⋅b|x∈11,13. Then, ℤ,+,∘ is a multiplicative hyperring. Consider the hyperideal expansion δ⋆ by δ⋆I=3ℤ+I. Since δ⋆2ℤ=3ℤ+2ℤ=ℤ, then, we conclude that I=2ℤ is a δ⋆-2-primary hyperideal of ℤ,+,∘.
Clearly, every prime hyperideal of a multiplicative hyperringR is a δ-2-primary hyperideal but its inverse is not true in general.Example 6.
Letℤ be the ring of integers, E be the set of all even integers of ℤ and A be the set of all positive even integers of ℤ. In the multiplicative hyperring ℤA (see Example 3.5 in [25]), the hyperideal E is δ1-2-primary but is not prime.Example 7.
In the multiplicative hyperringℤ,+,∘ with trivial hyperoperation, i.e., a∘b=ab for a,b∈ℤ, the hyperideal 2ℤ is δ1-2-primary.
We start the section with the following trivial result, and hence, we omit its proof.Theorem 12.
LetI be a proper of R. Then,(1)
I is a δ0-2-primary hyperideal if and only if I is a 2-prime hyperideal.(2)
IfI is a primary hyperideal, then, I is a δ1-2-primary hyperideal.(3)
IfI is a 2-prime hyperideal, then, I is a δ-2-primary hyperideal for every hyperideal expansion δ of R.(4)
IfI is a δ-primary hyperideal, then, I is a δ-2-primary hyperideal for every hyperideal expansion δ of R.(5)
IfI is a δ-2-primary hyperideal of R such that δI⊆γI for some hyperideal expansion γ of R, then, I is a γ-2-primary hyperideal of R.Theorem 13.
LetI be a hyperideal of a multiplicative hyperring R and δ a hypoerideal expansion of R. If I is a δ-2-primary hyperideal of R, then, for some idempotent element a∈R−I, I:a is a δ-2-primary hyperideal of R.Proof.
Letx∘y⊆I:a such that x2⊈I:a=I:a2 for some x,y∈R. This means that x°y°a⊆I but x2°a2⊈I. Since I is a δ-2-primary hyperideal of R, we get y2⊆δI⊆δI:a. Thus, I:a is a δ-2-primary hyperideal of R.Theorem 14.
LetI be a hyperideal of a multiplicative hyperring R such that for each a∈R−I, I:a=I:a2 and δ a hyperideal expansion of R. If the hyperideal I is irreducible, then, I is a δ-2-primary hyperideal of R.Proof.
We suppose thatI is not a δ-2-primary hyperideal of R and look for a contradiction. This means that there exist x,y∈R such that x∘y⊆I but x2⊈I and y2⊈δI. Thus, we get x∉I and y∉δI. Let t∈I+x∩I+y. Then, there are a1,a2∈I and r1,r2∈R such that for some t1∈r1∘x and t2∈r2∘y we have t=a1+t1=a2+t2. So, we get y∘a2+t2⊆y∘a2+y∘t2. Also, we have y∘a1+t1⊆y∘a1+y∘t1. Since y∘a1+y∘t1⊆I, then, y∘a2+t2⊆I. This implies that y∘a2+y∘t2∩I≠∅. Since I is a C-hyperideal of R, then, y∘a2+y∘t2⊆I. Hence, y∘t2⊆I which implies r2∘y2⊆I. Therefore, r2∈I:y2 which means r2∈I:y, by the assumption. So, r2∘y⊆I. Thus, we have t=a2+t2⊆a2+r2∘y⊆I. Then, I+x∩I+y⊆I. Since I⊆I+x∩I+y, then, we obtain I=I+x∩I+y. This is a contradiction since I is irreducible. Consequently, the hyperideal I of R is δ-2-primary.Theorem 15.
Letδ be a hyperideal expansion of a multiplicative hyperring R. Let I and δI be semiprime hyperideals of R. Then, I is a δ-2-primary hyperideal of R if and only if I is δ-primary.Theorem 16.
LetI be a hyperideal of a multiplicative hyperring R and δ a hyperideal expansion of R such that δI⊆δI. If I is a δ-2-primary hyperideal of R, then, I is a δ-primary hyperideal of R.Proof.
Letx∘y⊆I for some x,y∈R such that x∉I. This means that we have xn∘yn⊆I for some n∈ℕ. Clearly, x2n⊈I. Since I is a δ-2-primary hyperideal of R, then, y2n⊆δI which implies y∈δI⊆δI. This means that I is a δ-primary hyperideal of R.Theorem 17.
LetI be a hyperideal of a multiplicative hyperring R and δ a hyperideal expansion of R such that δI=δI. If the hyperideal I is primary, then, I is a δ-2-primary hyperideal of R.Proof.
Letx∘y⊆I for some x,y∈R. We have x∈I or y∈I⊆δI. Thus, x2⊆I or y2⊆δI=δI. Consequently, I is a δ-2-primary hyperideal of R.
Recall from [29] that a hyperideal expansion δ of R has the property of intersection preserving, if it satisfies δI∩J=δI∩δJ for any hyperideals I and J of R.Theorem 18.
Letδ has the property of intersection preserving. If Ii is a δ-2-primary hyperideal of R with δIi=P for all 1≤i≤n. Then, I=∩i=1nIi is a δ-2-primary hyperideal of R.Proof.
Leta,b∈I with a∘b⊆I and a2⊈I. Then, there exists some 1≤t≤n such that a2⊈It. Since It is a δ-2-primary hyperideal of R, then, b2⊆δIt=P. Therefore, b2⊆P=∩i=1nδIi=δ∩i=1nIi. Thus, we conclude that I=∩i=1nIi is a δ-2-primary hyperideal of R.Theorem 19.
IfIj|j∈Λ is a directed set of δ-2-primary hyperideals of a multiplicative hyperring R, then, ∪j∈ΛIj is a δ-2-primary hyperideal of R.Proof.
Letx∘y⊆∪j∈ΛIj for some x,y∈R. Assume that x2⊈∪j∈ΛIj. This implies that there exists i∈Λ such that x2⊈Ii. Since Ii is a δ-2-primary hyperideal of R, then, y2⊆δIi⊆δ∪j∈ΛIj. Thus, ∪j∈ΛIj is a δ-2-primary hyperideal of R.Theorem 20.
LetR be a regular multiplicative hyperring and δ be hyperideal expansion of R. If I is a δ-2-primary hyperideal of R, then, I is δ-primary.Proof.
LetI is a δ-2-primary hyperideal of R. Assume that a,b∈R and a∘b⊆I such that a∉I. Since R is regular, then, there exists r∈R such that a∈a2∘r. Let a2⊆I. Then, a∈a2∘r⊆I which is a contradiction. So, a2⊈I. Since I is a δ-2-primary hyperideal of R, then, b2⊆δI. Since R is regular, then, there exists r′∈R such that b∈b2∘r′⊆δI. Therefore, the hyperideal I is δ-primary.Theorem 21.
Letδ and γ be hyperideal expansions of R1 and R2, respectively, and f:R1⟶R2 a δγ-homomorphism. Then,(1)
IfI2 is a γ-2-primary hyperideal of R2, then, f−1I2 is a δ-2-primary hyperideal of R1.(2)
LetI1 be a hyperideal of R1 and f an epimorphism with Kerf⊆I1. Then, fI1 is a γ-2-primary hyperideal of R2 if and only if I1 is a δ-2-primary hyperideal of R1.Proof.
(1) Leta∘b⊆f−1I2 for some a,b∈R1. Then, we have fa∘b=fa∘fb⊆I2. Since I2 is a γ-2-primary hyperideal of R2, we get (fa)2⊆I2 or (fb)2⊆γI2 which implies fa2⊆I2 or fb2⊆γI2. Then, a2⊆f−1I2 or b2⊆f−1γI2=δf−1I2 because f is a δγ-homomorphism. Consequently, f−1I2 is a δ-2-primary hyperideal of R1. (2) ⇒ It is quite clear from (1).
⇐ Let a2∘b2⊆fI1 for some a2,b2∈R2. Then, for some a1,b1∈R1 we have fa1=a2 and fb1=b2. So, fa1∘fb1=fa1∘b1⊆fI1. Now, take any u∈a1∘b1. Then, fu∈fa1∘b1⊆fI1 and so there exists w∈I1 such that fu=fw. This means that fu−w=0, that is, u−w∈Kerf⊆I1 and then, u∈I1. Since I1 is a C-hyperideal of R1, then, we get a1∘b1⊆I1. Since I1 is a 2-primary hyperideal of R1, then, we obtain a12⊆I1 or b12⊆δI1. This implies that fa12=a22⊆fI1 or fb12=b22⊆fδI1=γfI1. Thus, fI1 is a γ-2-primary hyperideal of R2.Corollary 2.
LetI and J be two hyperideals of a multiplicative hyperring R such that I⊆J. Then, J is a δ-primary hyperideal of R if and only if J/I is a δq-primary hyperideal of R/I.Proof.
The claim is verified from Theorem21.Corollary 3.
LetI be a hyperideal of a multiplicative hyperring R and S a subhyperring of R with S⊈I. If I is a δ-2-primary hyperideal of R, then, I∩S is a δ-2-primary hyperideal of S.Theorem 22.
Letδ be a hyperideal expansion of R. Then, the followings are equivalent:(1)
Every proper principal hyperideal ofR is δ-2-primary.(2)
Every proper hyperideal ofR is δ-2-primary.Proof.
(1)⇒ (2) Assume that I is a proper hyperideal of R such that x∘y⊆I for some x,y∈R. Take any t∈x∘y. Clearly, x∘y∩t≠∅. This implies that x∘y⊆t⊆I. Since t is a δ-2-primary hyperideal of R, then, we get x2⊆t⊆I or y2⊆δt⊆δI. Therefore, I is a δ-2-primary hyperideal of R. 2⇒1 It is clear.
LetR1,+1,∘1 and R2,+2,∘2 be two multiplicative hyperrings with nonzero identity [29]. Recall R1×R2,+,∘ is a multiplicative hyperring with the operation + and the hyperoperation ∘ are defined respectively asx1,x2+y1,y2=x1+1y1,x2+2y2 andx1,x2∘y1,y2=x,y∈R1×R2|x∈x1∘1y1,y∈x2∘2y2.
Assume thatδ1 and δ2 are hyperideal expansions of R1 and R2, respectively. If δR1×R2 is a function of hyperideals of R with δR1×R2I1×I2=δ1I1×δ2I2 for every hyperideals I1 and I2 of R1 and R2, respectively, then, δR1×R2 is a hyperideal expansion of R1×R2.Theorem 23.
LetR1,+1,∘1 and R2,+2,∘2 be two multiplicative hyperrings with nonzero identity such that δ1 and δ2 be hyperideal expansions of R1 and R2, respectively. Let I1 be a hyperideal of R1. Then, the hyperideal I1 of R1 is δ1-2-primary if and only if I1×R2 is a δR1×R2-2-primary of R1×R2.Proof.
⇒ Let x1,x2∘y1,y2⊆I1×R1 for some x1,x2,y1,y2∈R1×R2. This means x1∘1y1⊆I1. Since I1 is a δ1-2-primary hyperideal of R1, then, we get x12⊆I1 or y12⊆δ1I1. This implies that x1,x22=x12,x22⊆I1×R2 or y1,y22=y12,y22⊆δR1×R2I1×R2.
⇐ Assume on the contrary that I1 is not a δ1-2-primary hyperideal of R1. So, x1∘1y1⊆I1 with x1,y1∈R1 implies that x12⊈I1 and y12⊈δ1I1. It is clear that x1,1R2∘y1,1R2⊆I1×R2. Since I1×R2 is a δR1×R2-2-primary hyperideal of R1×R2, then, we have x1,1R22=x12,1R2⊆I1×R2 or y1,1R22=y12,1R2⊆δR1×R2I1×R2. Hence, we get x12⊆I1 or y12⊆δ1I1 which is a contradiction. Consequently, I1 is a δ1-2-primary hyperideal of R1.Theorem 24.
LetR1,+1,∘1 and R2,+2,∘2 be two multiplicative hyperrings with nonzero identity such that δ1 and δ2 be hyperideal expansions of R1 and R2, respectively. Let I1 and I2 be some hyperideals of R1 and R2, respectively. Then, the following statements are equivalent:(1)
I1×I2 is a δR1×R2-2-primary hyperideal of R1×R2.(2)
I1=R1 and I2 is a δ2-2-primary hyperideal of R2 or I2=R2 and I1 is a δ1-2-primary hyperideal of R1.Proof.
(1)⇒ (2) Assume that I1=R1. Then, I2 is a δ2-2-primary hyperideal of R2, by Theorem 23.
(2)⇒ (1) This can be proved, by using Theorem 23.Example 8.
Suppose thatℤ,+,⋅ is the ring of integers. Then, ℤ,+,∘1 is a multiplicative hyperring with a hyperoperation a∘1b=ab,7ab. Also, ℤ,+,∘2 is a multiplicative hyperring with a hyperoperation a∘2b=ab,5ab. Note that ℤ×ℤ,+,∘ is a multiplicative hyperring with a hyperoperation a,b∘c,d=x,y∈ℤ×ℤ|x∈a∘1c,y∈b∘2d. Clearly, 7ℤ=7t|t∈ℤ and 5ℤ=5t|t∈ℤ are two δ0-2-primary of ℤ,+,∘1 and ℤ,+,∘2, respectively. Since 5,0∘0,7⊆7ℤ×5ℤ, but 5,02,0,72⊈7ℤ×5ℤ and 5,02,0,72⊈δℤ×ℤ7ℤ×5ℤ=7ℤ×5ℤ, then, 7ℤ×5ℤ is not a δ0×δ0-2-primary hyperideal of ℤ×ℤ.



---

*Source: 5721946-2022-03-27.xml*