# Note on Isomorphism Theorems of Hyperrings

Authors: Muthusamy Velrajan; Arjunan Asokkumar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

There are different notions of hyperrings(R,+,·). In this paper, we
extend the isomorphism theorems to hyperrings, where the additions and the multiplications
are hyperoperations.


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## Body

## 1. Introduction

The theory of hyperstructures was introduced in 1934 by Marty [1] at the 8th Congress of Scandinavian Mathematicians. This theory has been subsequently developed by Corsini [2–4], Mittas [5, 6], Stratigopoulos [7], and by various authors. Basic definitions and propositions about the hyperstructures are found in [3, 4, 8]. Krasner [9] has studied the notion of hyperfields, hyperrings, and then some researchers, namely, Ameri [10], Dašić [11], Davvaz [12], Gontineac [13], Massouros [14], Pianskool et al. [15], Sen and Dasgupta [16], Vougiouklis [8, 17], and others followed him.Hyperrings are essentially rings with approximately modified axioms. There are different notions of hyperrings(R,+,·). If the addition + is a hyperoperation and the multiplication · is a binary operation, then the hyperring is called Krasner (additive) hyperring [9]. Rota [18] introduced a multiplicative hyperring, where + is a binary operation and the multiplication · is a hyperoperation. De Salvo [19] studied hyperrings in which the additions and the multiplications were hyperoperations. These hyperrings were also studied by Barghi [20] and by Asokkumar and Velrajan [21–23]. In 2007, Davvaz and Leoreanu-Fotea [24] published a book titled Hyperring Theory and Applications. Davvaz [12] extended that the isomorphism theorems to Krasner hyperrings, provided the hyperideals considered in the isomorphism theorems are normal.In this paper, we extend the isomorphism theorems to hyperrings, in which both the additions and the multiplications are hyperoperations. Also, it is observed that ifI is an hyperideal of a hyperring R and (I,+) is a normal subcanonical hypergroup of (R,+), then R/I is a ring, and hence the quotient hyperrings considered in the isomorphism theorems by Davvaz in [12] are rings.


## 2. Basic Definitions and Notations

This section provides some basic definitions that have been used in the sequel. Ahyperoperation∘ on a nonempty set H is a mapping of H×H into the family of nonempty subsets of H (i.e., x∘y⊆H, for every x,y∈H). The definitions are found in references [3, 4, 8, 24]. A hypergroupoid is a nonempty set H equipped with a hyperoperation ∘. For any two nonempty subsets A and B of a hypergroupoid H and for x∈H, A∘B, we mean the set ⋃a∈Ab∈B(a∘b), A∘x=A∘{x}, and x∘B={x}∘B.A hypergroupoid(H,∘) is called a semihypergroup if x∘(y∘z)=(x∘y)∘z for every x,y,z∈H (the associative axiom). A hypergroupoid (H,∘) is called a quasihypergroup if x∘H=H∘x=H for every x∈H (the reproductive axiom). A reproductive semihypergroup is called a hypergroup (in the sense of Marty). A comprehensive review of the theory of hypergroups appears in [3].Definition 2.1.
A nonempty setH with a hyperoperation + is said to be a canonical  hypergroup if the following conditions hold: (i)
for everyx,y∈H, x+y=y+x,(ii)
for everyx,y,z∈H, x+(y+z)=(x+y)+z,(iii)
there exists0∈H (called neutral element of H) such that 0+x={x}=x+0 for all x∈H,(iv)
for everyx∈H, there exists a unique element denoted by -x∈H such that 0∈x+(-x)∩(-x)+x,(v)
for everyx,y,z∈H, z∈x+y implies y∈-x+z and x∈z-y.Example 2.2.
Consider the setH={0,x,y}. Define a hyperaddition + on H as in the following table. Then, (H,+) is a canonical hypergroup. 							(2.1)+0xy00xyxx{0,x}yyyy{0,x,y}The following elementary facts in a canonical hypergroupH easily follow from the axioms.(i)
-(-a)=a for every a∈H,(ii)
0 is the unique element such that for everya∈H, there is an element -a∈H with the property 0∈a+(-a),(iii)
-0=0,(iv)
-(a+b)=-a-b for all a,b∈H.For any subsetA of a canonical hypergroup H, -A denotes the set {-a:a∈A}. A nonempty subset N of a canonical hypergroup of H is called a subcanonical  hypergroup of H if N is a canonical hypergroup under the same hyperoperation as that of H. Equivalently, for every x,y∈N, x-y⊆N. In particular, for any x∈N, x-x⊆N. Since 0∈x-x, it follows that 0∈N.Definition 2.3.
An equivalence relationρ defined on a canonical hypergroup (H,+) is called strongly  regular if for all x,y∈H and xρy implies that for every p∈H, for every a∈x+p and for every b∈y+p one has aρb.Definition 2.4.
A subcanonical hypergroupA of a canonical hypergroup H is said to be normal if x+A-x⊆A for all x∈A.Definition 2.5.
Theheart of a canonical hypergroup H is the union of the sums (x1-x1)+(x2-x2)+(x3-x3)+⋯+(xn-xn), where xi∈H and n is a natural number and it is denoted by ωH.Definition 2.6.
LetH1 and H2 be two canonical hypergroups. A mapping ϕ from H1 into H2 is called a homomorphism from H1 into H2 if (i) ϕ(a+b)⊆ϕ(a)+ϕ(b) for all a,b∈H1 and (ii) ϕ(0)=0 hold. The mapping ϕ is called a good  or  strong  homomorphism if (i) ϕ(a+b)=ϕ(a)+ϕ(b) for all a,b∈H1 and (ii) ϕ(0)=0 hold.A homomorphism (resp., strong homomorphism)ϕ from a canonical hypergroup H1 to a canonical hypergroup H2 is called an isomorphism  (resp.,  strong  isomorphism) if ϕ is one to one and onto. If H1 is strongly isomorphic to H2, then we denote it by H1≅H2.Definition 2.7.
Letϕ be a homomorphism from canonical hypergroup H1 into a canonical hypergroup H2. Then, the set {x∈H1:ϕ(x)=0} is called kernel of ϕ and is denoted by Kerϕ, and the set {ϕ(x):x∈H1} is called Image of ϕ and is denoted by Imϕ.It is clear thatKerϕ is a subcanonical hypergroup of H1 and Imϕ is a subcanonical hypergroup of H2. The definition of a hyperring given below is equivalent to one formulated by De Salvo [19] (see Corsini [3]) and studied by Barghi [20].Definition 2.8.
Ahyperring is a triple (R,+,·), where R is a nonempty set with a hyperaddition + and a hypermultiplication · satisfying the following axioms: (1)
(R,+) is a canonical hypergroup,(2)
(R,·) is a semihypergroup such that x·0=0·x=0 for all x∈R, (i.e, 0 is a bilaterally absorbing element),(3)
The hypermultiplication· is distributive with respect to the hyperoperation +. That is, for every x,y,z∈R, x·(y+z)=x·y+x·z and (x+y)·z=x·z+y·z.
In a hyperring if the hypermultiplication is a binary operation, then it is called asKrasner  or  additive  hyperring. Also, in the Definition 2.8, if the hyperaddition is a binary operation, then it is called as multiplicative  hyperring.Example 2.9.
LetR={0,1} be a set with two hyperoperations defined as follows:
(2.2)+010{0}{1}1{1}{0,1}⋅010{0}{0}1{0}{0,1}Then,(R,+,·) is a hyperring.Definition 2.10.
LetR be a hyperring, and let I be a nonempty subset of R. I is called a left (resp., right) hyperideal of R if (I,+) is a canonical subhypergroup of R and for every a∈I and r∈R, ra⊆I (resp., ar⊆I). A hyperideal of R is one which is a left as well as a right hyperideal of R.IfI,J are left (resp., right) hyperideals of a hyperring R, then I+J, I∩J are left (resp., right) hyperideal of R. If I,J are hyperideals of a hyperring R, then I+J, I∩J are hyperideals of R.Definition 2.11.
LetR1 and R2 be two hyperrings. A mapping ϕ from R1 into R2 is called a homomorphism if (i) ϕ(a+b)⊆ϕ(a)+ϕ(b); (ii) ϕ(ab)⊆ϕ(a)ϕ(b) and (iii) ϕ(0)=0 hold for all a,b∈R1. The mapping ϕ is called a good  homomorphism or a  strong  homomorphism if (i) ϕ(a+b)=ϕ(a)+ϕ(b); (ii) ϕ(ab)=ϕ(a)ϕ(b) and (iii) ϕ(0)=0 hold for all a,b∈R1.Definition 2.12.
A homomorphism (resp., strong homomorphism)ϕ from hyperring R1 into a hyperring R2 is said to be an isomorphism  (resp., strong isomorphism) if ϕ is one to one and onto. If R1 is strongly isomorphic to R2, then it is denoted by R1≅R2.Remark 2.13.
Letϕ be a homomorphism from a hyperring R1 into a hyperring R2. Then Kerϕ is a hyperideal of R1 and Imϕ is a hyperideal of R2.


## 3. Canonical Hypergroups

LetN be a subcanonical hypergroup of a canonical hypergroup H. In this section, we construct quotient canonical hypergroup H/N and prove that when N is normal, H/N is an abelian group.Proposition 3.1.
LetH be a canonical hypergroup, and let N be a subcanonical hypergroup of H. For any two elements a,b∈H, if we define a relation a~b if a∈b+N, then ~ is an equivalence relation on H.Proof.
Leta∈H. Since a=a+0∈a+N, the relation ~ is reflexive. Let a,b∈H. If a∈b+N, then a∈b+n for some n∈N. That is, b∈a-n⊆a+N. So, ~ is a symmetric relation. Suppose that a,b,c∈H such that a~b and b~c, then a∈b+N and b∈c+N. Therefore, a∈b+n, and b∈c+m, for some n,m∈N. So, a∈c+m+n⊆c+N. Hence a~c. Therefore, the relation ~ is transitive.Remark 3.2.
LetN be a subcanonical hypergroup of a canonical hypergroup H. We denote the equivalence class determined by the element x∈H by the equivalence relation ~ by x¯. It is clear that x¯=x+N.Proposition 3.3.
LetH be a canonical hypergroup, and let N be a normal subcanonical hypergroup of H. Then, for x,y∈N, the following are equivalent:(1)
y∈x+N,(2)
x-y⊆N,(3)
(x-y)∩N≠∅.Proof.
(1) implies (2).
Sincey∈x+N, we have y-x⊆x+N-x. Since N is normal subcanonical hypergroup of H, we get x+N-x⊆N. Thus, y-x⊆N. That is, -(y-x)⊆N, and hence x-y⊆N.
(2) implies (3) is obvious.
(3) implies  (1). Since(x-y)∩N≠∅, there exists a∈x-y and a∈N. Therefore, -y+x⊆-y+a+y⊆N. If z∈-y+x, then z∈N. Therefore, -y∈z-x. That is, y∈x-z⊆x+N.Remark 3.4.
LetH be a canonical hypergroup, and let N be a subcanonical hypergroup of H. When N is normal, the equivalence relation defined in the Proposition 3.1 coincides with the the equivalence relation defined by Davvaz [12]. Further, the Propositions 3.1 and 3.3 are true when the hyperaddition on the canonical hypergroup H is not commutative. Also, for any x∈H, we have (-x¯)=-(x¯).Theorem 3.5.
LetH be a canonical hypergroup, N be a subcanonical hypergroup of H. Then for x,y∈H, the sets A={z¯:z∈x+y}, B={z¯:z∈x¯+y¯} and C={z¯:z¯⊆x¯+y¯} are equal.Proof.
Letz¯∈A. Then z∈x+y. Since x∈x¯ and y∈y¯ we have z∈x¯+y¯. Thus A⊆B. Suppose z¯∈B, then z∈x¯+y¯. That is, z∈t+n for some t∈x+y and n∈N. Therefore z¯=t¯, where t∈x+y. Since t¯∈A, we get z¯∈A. Thus B⊆A. Hence A=B.
Ifz¯∈A, then z∈x+y. Therefore, z¯⊆x+y+N=x+N+y+N=x¯+y¯. Hence A⊆C. On the other hand if z¯∈C, then z¯⊆x¯+y¯. Since z∈z¯⊆x¯+y¯, we get z∈s+n for some s∈x+y and n∈N. Thus z¯=s¯. Since s¯∈A, we get C⊆A. Hence A=C.Remark 3.6.
LetH be a canonical hypergroup, and let N be a subcanonical hypergroup of H. Then, we denote the collection of all equivalence classes {x¯:x∈H} induced by the equivalence relation ~ by H/N.Theorem 3.7.
LetH be a canonical hypergroup, and let N be a subcanonical hypergroup of H. If we define x¯⊕y¯={z¯:z∈x+y} for all x¯,y¯∈H/N, then H/N is a canonical hypergroup.Proof.
Ifx1,y1,x2,y2∈H such that x1¯=x2¯ and y1¯=y2¯, then x2∈x1+N and y2∈y1+N. Let z2∈x2+y2⊆(x1+N)+(y1+N). Since H is commutative, z2∈z1+i for some z1∈x1+y1 and for some i∈N. That is, z2+N=z1+N. Hence, x2¯⊕y2¯⊆x1¯⊕y1¯. Also, since x1∈x2+N and y1∈y2+N, by a similar argument, we get, x1¯⊕y1¯⊆x2¯⊕y2¯. Hence, x1¯⊕y1¯=x2¯⊕y2¯. Thus, hyperaddition ⊕ is well defined.
Letx¯,y¯,z¯∈H/N. If u¯∈(x¯⊕y¯)⊕z¯, then u¯∈p¯⊕z¯ for some p¯∈x¯⊕y¯. That is, u¯=a¯ for some a∈p+z. Also, p¯=b¯ for some b∈x+y. Now, a∈p+z⊆b+N+z=b+z+N. That is, a∈v+N for some v∈b+z⊆(x+y)+z=x+(y+z). So, v∈x+t for some t∈y+z. This means that a¯=v¯ and v¯∈x¯⊕t¯. Since t¯∈y¯⊕z¯, we have u¯=a¯=v¯∈x¯⊕t¯⊆x¯⊕(y¯⊕z¯). This means that u¯∈x¯⊕(y¯⊕z¯). Hence (x¯⊕y¯)⊕z¯⊆x¯⊕(y¯⊕z¯). Similarly, we get x¯⊕(y¯⊕z¯)⊆(x¯⊕y¯)⊕z¯. Hence, x¯⊕(y¯⊕z¯)=(x¯⊕y¯)⊕z¯. Thus, the hyperaddition is associative.
Consider the element0¯=0+N∈H/N. Now, for any x∈H, we have x¯⊕0¯={z¯:z∈x+0}=x¯. Similarly, 0¯⊕x¯=x¯. Thus, 0¯ is the zero element of H/N.
Letx∈H, then x¯⊕(-x¯)={z¯:z∈x+(-x)=x-x}. Since 0∈x-x, we get 0¯∈x¯⊕(-x¯). Similarly, we can show that 0¯∈(-x¯)⊕x¯. Let x¯∈H/N, and suppose that y¯∈H/N is such that 0¯∈y¯⊕x¯, then 0¯=a¯, where a∈y+x. That is, y∈a-x⊆N-x, and hence y¯=-x¯. Thus, the element x¯∈H/N has a unique inverse -x¯∈H/N.
Suppose thatz¯∈x¯⊕y¯, then z¯=a¯, where a∈x+y. This implies x∈a-y⊆z+N-y. That is, x∈r+N, where r∈z-y. Thus, x¯=r¯∈z¯⊕(-y¯). Similarly, we can show y¯∈(-x¯)⊕z¯. Since H is commutative, it is obvious that H/N is also commutative. Thus, H/N is a canonical hypergroup.Corollary 3.8.
Letϕ be a strong homomorphism from canonical hypergroup H1 into a canonical hypergroup H2, then H1/Kerϕ is a canonical hypergroup.Remark 3.9.
LetH be a canonical hypergroup, and let A be a subcanonical hypergroup of H. We denote the subset {x∈H:x-x⊆A} of H by SA.Proposition 3.10.
LetH be a canonical hypergroup, and let A be a subcanonical hypergroup of H. Then, SA is a subcanonical hypergroup of H containing A.Proof.
Letx∈A. Since A is a subcanonical hypergroup of H, x-x⊆A. This implies x∈SA. Therefore, A⊆SA. Since A≠∅, the set SA is nonempty.
Letx,y∈SA. For r∈x-y, we get r-r⊆(x-y)-(x-y)=(x-x)+(y-y)⊆A+A=A. Hence, r∈SA. That is, x-y⊆SA. Therefore, SA is a subcanonical hypergroup of H containing A.Definition 3.11.
Let(H,+) be a canonical hypergroup, and let A be a subcanonical hypergroup of H.  A is called a subgroup of H if (A,+) is a group. That is, x+y is a singleton set for all x,y∈A.Example 3.12.
The setH={0,a,b,c} with the following hyperoperation + is a canonical hypergroup
(3.1)+0abc0{0}{a}{b}{c}a{a}{0,b}{a,c}{b}b{b}{a,c}{0,b}{a}c{c}{b}{a}{0}In this example{0,c},{0} are subgroups of H and ωH={0,b} whereas in the Example 2.2, {0} is the subgroup of H and ωH=H.Proposition 3.13.
LetH be a canonical hypergroup. Then, S{0} is the subgroup of H containing all subgroups of H.Proof.
By the Proposition3.10, S{0} is the subcanonical hypergroup of H. Let x,y∈S{0}. Consider the set x+y. If u,v∈x+y, then u-v⊆(x+y)-(x+y)=(x-x)+(y-y)=0+0=0. Hence, u=v. This means that the set x+y has only one element. Thus, S{0} is a subgroup of H. Suppose, A is any subgroup of H, then for any x∈A that we have x-x=0. That is, x∈S{0}. Hence, A⊆S{0}. Thus, S{0} contains all subgroups of H.Corollary 3.14.
LetH be a canonical hypergroup. Then, H is an abelian group if and only if S{0}=H.Proposition 3.15.
LetH be a canonical hypergroup, and let A be a subcanonical hypergroup of H. Then, A is normal if and only if SA=H.Proof.
LetA be normal. Then, for x∈H, x+0-x⊆A. That is, x∈SA. Hence, SA=H. Conversely, if SA=H, then for x∈H, we get x+A-x=x-x+A⊆A+A=A. Thus, A is normal.Proposition 3.16.
The heartωH of a canonical hypergroup H is a normal subcanonical hypergroup of H.Proof.
Ifx,y∈ωH, then x∈(x1-x1)+(x2-x2)+(x3-x3)+⋯+(xn-xn) and y∈(y1-y1)+(y2-y2)+(y3-y3)+⋯+(ym-ym), where xi,yj∈H and m,n are natural numbers. Thus x-y∈(x1-x1)+(x2-x2)+(x3-x3)+⋯+(xn-xn)+(y1-y1)+(y2-y2)+(y3-y3)+⋯+(ym-ym)⊆ωH. Now, for any element h∈ωH, there exists natural number n and elements xi∈H such that h∈(x1-x1)+(x2-x2)+(x3-x3)+⋯+(xn-xn). Then, for any x∈H, x+h-x=x-x+h⊆x-x+(x1-x1)+(x2-x2)+(x3-x3)+⋯+(xn-xn)⊆ωH. Hence, heart ωH is a normal subcanonical hypergroup of H.Proposition 3.17.
A subcanonical hypergroupA of a canonical hypergroup H is normal if and only if A contains the heart ωH of the canonical hypergroup H.Proof.
LetA be a normal subcanonical hypergroup of the canonical hypergroup H. Then x+i-x⊆A for every x∈H, and i∈A. In particular, when i=0∈A, we get x-x⊆A for every x∈H. Since A is a subcanonical hypergroup of H, the union of the sums (x1-x1)+(x2-x2)+(x3-x3)+···+(xn-xn)⊆A for xi∈H and n is a natural number. That is, ωH⊆A. Conversely, assume that subcanonical hypergroup A contains the heart ωH of the canonical hypergroup H. For x∈H and i∈A, x+i-x=x-x+i⊆ωH+A⊆A+A=A. Hence, A is a normal subcanonical hypergroup.From Propositions3.16 and 3.17, we have the following proposition.Proposition 3.18.
In a canonical hypergroupH, ωH is the smallest normal subcanonical hypergroup.Proposition 3.19.
LetA,B be subcanonical hypergroups of a canonical hypergroup H such that A⊆B, then SA⊆SB.Proof.
Letx∈SA. Then, x-x⊆A. That is, x∈SB. Hence, SA⊆SB.Proposition 3.20.
LetA,B be subcanonical hypergroups of a canonical hypergroup H such that A⊆B. If A is normal, then B is also normal.Proof.
IfA is normal, then by Proposition 3.15, SA=H. Since A⊆B, by Proposition 3.19, SA⊆SB. Hence, H=SB. By Proposition 3.15, B is normal.Corollary 3.21.
LetA,B be subcanonical hypergroups of a canonical hypergroup H such that A is normal, then the subcanonical hypergroup A+B is also normal.Corollary 3.22.
LetH be a canonical hypergroup such that (0) is normal, then all the subcanonical hypergroups are normal.Theorem 3.23.
LetH be a canonical hypergroup. Then, the following are equivalent:(i)
H is an abelian group,(ii)
(0) is a normal subcanonical hypergroup of H,(iii)
ωH=(0).Proof.
By Corollary3.14, a canonical hypergroup H is an abelian group if and only if S{0}=H. By Proposition 3.15, S{0}=H if and only if (0) is a normal subcanonical hypergroup of H. Hence, a canonical hypergroup H is an abelian group if and only if (0) is a normal subcanonical hypergroup of H.
By Proposition3.18, ωH is the smallest normal subcanonical hypergroup of H. Therefore, (0) is normal if and only if ωH=(0).Corollary 3.24.
H is an abelian group if and only if all subcanonical hypergroups of H are normal.Theorem 3.25.
LetH be a canonical hypergroup, and let N be a normal subcanonical hypergroup of H. Then, H/N is an abelian group.Proof.
For the quotient canonical hypergroupH/N, the zero element is N. Since (x+N)+N+(-x+N)=(x+N-x)+N⊆N+N=N for all x∈H, we have {N} is a normal subcanonical hypergroup in H/N. By Theorem 3.23, H/N is an abelian group.Remark 3.26.
IfN is a normal subcanonical hypergroup of a canonical hypergroup H, then the relation ~ defined in Proposition 3.1, is a strongly regular equivalence relation. Hence, by Theorem  31 in [3], H/N is an abelian group. However, we have proved Theorem 3.25 in a different way.


## 4. Isomorphism Theorems of Canonical Hypergroups

In this section, we prove the isomorphism theorems of canonical hypergroups.Theorem 4.1 (First Isomorphism Theorem).
Letϕ be a strong homomorphism from a canonical hypergroup H1 into a canonical hypergroup H2 with kernel K. Then, H1/K is strongly isomorphic to Imϕ.Proof.
Define a mapf:H1/K→Imϕ by f(x¯)=ϕ(x) for all x∈H1. Suppose that x¯=y¯, where x,y∈H, then x∈y¯. That is, x∈y+k for some k∈K. Hence, ϕ(x)∈ϕ(y+k)=ϕ(y)+ϕ(k)=ϕ(y)+0=ϕ(y). So ϕ(x)=ϕ(y). Hence, f(x¯)=f(y¯). Thus, the map f is well defined.
Ifx,y∈H1, then
(4.1)f(x¯⊕y¯)=f({z¯:z∈x+y})={f(z¯):z∈x+y}={ϕ(z):z∈x+y}.Also,(4.2)f(x¯)+f(y¯)=ϕ(x)+ϕ(y)=ϕ(x+y)={ϕ(z):z∈x+y}.
Thus, f(x¯⊕y¯)=f(x¯)+f(y¯). Moreover, f(0¯)=ϕ(0)=0. Hence, f is a strong homomorphism.
Suppose thatx¯,y¯∈H1/K such that f(x¯)=f(y¯), then ϕ(x)=ϕ(y). This means that 0∈ϕ(x)-ϕ(y)=ϕ(x-y). That is, ϕ(z)=0 for some z∈x-y. Since ϕ(z)=0, we get z∈K. Now, z∈x-y⇒x∈z+y⇒x∈y+K. Then, by Proposition 3.3  x¯=y¯ and hence f is one to one. Clearly, f is onto. Thus, f is a strong isomorphism. That is, H1/K is strongly isomorphic to Imϕ.Corollary 4.2.
Letϕ be a strong homomorphism from a canonical hypergroup H1 onto a canonical hypergroup H2 with kernel K. Then, H1/K is isomorphic to H2.Theorem 4.3 (Second Isomorphism Theorem).
IfM and N are subcanonical hypergroups of a canonical hypergroup H, then N/(M∩N)≅(M+N)/M.Proof.
It is clear that we can consider the subcanonical hypergroupM+N of the canonical hypergroup H as a canonical hypergroup M+N for which M is a subcanonical hypergroup. Similarly, the subcanonical hypergroup N of the canonical hypergroup H as a canonical hypergroup N for which (M∩N) is a subcanonical hypergroup.
Defineg:N→(M+N)/M by g(b)=b+M for every b∈N. For all a,b∈N, g(a+b)=g({x:x∈a+b})={g(x):x∈a+b}={x+M:x∈a+b}=(a+M)⊕(b+M)=g(a)⊕g(b). Moreover, g(0)=0. Thus, g is a strong homomorphism.
Now,x+M∈(M+N)/M implies that x∈y+M for some y∈M+N. That is, y∈a+b for some a∈M, b∈N. Since y∈b+M, we get y+M=b+M. Thus, g(b)=b+M=y+M=x+M. Thus, g is onto. Let b∈N. Then, b∈Kerg⇔g(b)=0⇔b+M=0+M⇔b∈M. Thus, b∈Kerg if and only if b∈M∩N. Hence, by the First Isomorphism Theorem, N/(M∩N)≅(M+N)/M.Theorem 4.4 (Third Isomorphism Theorem).
IfM and N are subcanonical hypergroup of a canonical hypergroup H such that M⊆N, then H/N≅(H/M)/(N/M).Proof.
Define a maph:H/N→H/M by h(x+N)=x+M. Then, h is a strong onto homomorphism of canonical hypergroup with kernel N/M. Therefore, by the First Isomorphism Theorem of canonical hypergroups, H/N≅(H/M)/(N/M).


## 5. Isomorphism Theorems of Hyperrings

LetR be a hyperring, and let I be a hyperideal of R. Since I is a subcanonical hypergroup of R,R/I={x¯:x∈R} is a canonical hypergroup under the hyperaddition defined in the Theorem 3.7. In this section, we define a hypermultiplication on R/I and prove that R/I is a hyperring.Theorem 5.1.
If we definex¯⊗y¯={z¯:z∈xy} for all x¯,y¯∈R/I, then R/I is a hyperring.Proof.
Ifx1,y1,x2,y2∈R such that x1¯=x2¯ and y1¯=y2¯, then x2∈x1+I and y2∈y1+I. Let z2∈x2y2⊆(x1+I)(y1+I)⊆x1y1+I. Then, z2∈z1+i for some z1∈x1y1 and for some i∈I. That is, z2+I=z1+I and so x2¯⊗y2¯⊆x1¯⊗y1¯. Similarly, we get, x1¯⊗y1¯⊆x2¯⊕y2¯. Hence, x1¯⊕y1¯=x2¯⊗y2¯. Thus, hypermultiplication ⊗ is well defined.
Suppose,x¯,y¯,z¯∈R/I. Then,
(5.1)x¯⊗(y¯⊗z¯)=x¯⊗{a¯:a∈yz}={s¯:s∈xa,a∈yz}={s¯:s∈x(yz)}={s¯:s∈(xy)z}={s¯:s∈bz,b∈xy}={s¯:s∈xy}⊗z¯=(x¯⊗y¯)⊗z¯
Thus, we get x¯⊗(y¯⊗z¯)=(x¯⊗y¯)⊗z¯. Hence, hypermultiplication is associative. Further,
(5.2)x¯⊗(y¯⊕z¯)=x¯⊗{p+I:p∈y+z}={q+I:q∈xp,p∈y+z}={q+I:q∈x(y+z)}={q+I:q∈xy+xz}.
Also,
(5.3)(x¯⊗y¯)⊕(x¯⊗z¯)={a+I:a∈xy}⊕{b+I:b∈xz}={c+I:c∈a+b,a∈xy,b∈xz}={c+I:c∈xy+xz}.
Hence, x¯⊗(y¯⊕z¯)=(x¯⊗y¯)⊕(x¯⊗z¯). Similarly, we can show that (x¯⊕y¯)⊗z¯=(x¯⊗z¯)⊕(x¯⊗z¯). Therefore, hypermultiplication is distributive with respect to the hyperaddition. Thus, R/I is a hyperring.Corollary 5.2.
Letϕ be a strong homomorphism from hyperring R1 into a hyperring R2, then R1/Kerϕ is a hyperring.Remark 5.3.
IfR is a Krasner hyperring and I is a hyperideal of R, then R/I is also a Krasner hyperring. Further if (I,+) is a normal subcanonical hypergroup of R, then by the Theorems 3.23 and 5.1, R/I is a ring. Hence, the quotient hyperrings considered in [12] are just rings. So, in the isomorphism theorems proved in [12], all the quotient hyperrings considered are rings. However, we prove the isomorphism theorems of hyperrings in which the additions and the multiplications are hyperoperations.
IfR is a hyperring, and I is a hyperideal of R, and (I,+) is a normal subcanonical hypergroup of R, then R/I is a multiplicative hyperring.Theorem 5.4 (First Isomorphism Theorem).
Letϕ be a strong homomorphism from a hyperring R1 into a hyperring R2 with kernel K. Then, R1/K is strongly isomorphic to Imϕ.Proof.
Define a mapf:R1/K→Imϕ by f(x¯)=ϕ(x) for all x∈R1.
By Theorem4.1, this map f is a strong isomorphism from canonical hypergroup R1/K onto Imϕ. Now,
(5.4)f(x¯⊗y¯)=f({z¯:z∈xy})={f(z¯):z∈xy}={ϕ(z):z∈xy},f(x¯)f(y¯)=ϕ(x)ϕ(y)=ϕ(xy)={ϕ(z):z∈xy}.
Thus,f(x¯⊗y¯)=f(x¯)f(y¯). Hence, f is a strong hyperring isomorphism.Corollary 5.5.
Letϕ be a strong homomorphism from a hyperring R1 onto a hyperring R2 with kernel K. Then, R1/K is strongly isomorphic to R2.Theorem 5.6 (Second Isomorphism Theorem).
IfI and J are hyperideals of a hyperring R, then J/(I∩J)≅(I+J)/I.Proof.
We can consider the hyperidealI+J of the hyperring R as a hyperring I+J for which I is a hyperideal. Similarly, hyperideal J of the hyperring R as a hyperring J for which (I∩J) is a hyperideal.
Defineg:J→(I+J)/I by g(b)=b+I for every b∈J. By  Theorem 4.3,g is strong isomorphism from canonical hypergroup J onto the canonical hypergroup (I+J)/I. Now, g(ab)=g({x:x∈ab})={g(x):x∈ab}={x+I:x∈ab}=(a+I)(b+I)=g(a)g(b). Thus, g is strong isomorphism from hyperring J onto the hyperring (I+J)/I. Also, from  Theorem 4.3,Kerg=I∩J. Hence, by First Isomorphism Theorem of hyperrings, J/(I∩J)≅(I+J)/I.Theorem 5.7 (Third Isomorphism Theorem).
IfI and J are hyperideals of a hyperring R such that I⊆J, then R/J≅(R/I)/(J/I).Proof.
Define a maph:R/J→R/M by h(x+I)=x+J. Then, h is a strong onto homomorphism of hyperring with kernel J/I. Therefore, by the First Isomorphism Theorem of hyperrings, R/J≅(R/I)/(J/I).



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