# Associate, Hyperdomainlike, and Presimplifiable Hyperrings

Authors: Agboola Adesina Abdul Akeem; Davvaz Bijan
Journal: Journal of Mathematics
 (2014)

---

## Abstract

Based on the works of Axtell et al., Anderson et al., and Ghanem on associate, domainlike, and presimplifiable rings, we
introduce new hyperrings called associate, hyperdomainlike, and presimplifiable hyperrings. Some elementary properties of these new hyperrings and their relationships are presented.


---

## Body

## 1. Introduction

The study of strongly associate rings began with Kaplansky in [1] and was further studied in [2–5]. Domainlike rings and their properties were presented by Axtell et al. in [6]. Presimplifiable rings were introduced by Bouvier in the series of papers [7–11] and were later studied in [2–4]. Further properties of associate and presimplifiable rings were recently presented by Ghanem in [12].The theory of hyperstructures was introduced in 1934 by Marty [13] at the 8th Congress of Scandinavian Mathematicians. Introduction of the theory has caught the attention and interest of many mathematicians and the theory is now spreading like wild fire. The notion of canonical hypergroups was introduced by Mittas [14]. Some further contributions to the theory can be found in [15–19].Hyperrings are essentially rings with approximately modified axioms. Hyperrings(
R
,
+
,
·
) are of different types introduced by different researchers. Krasner [20] introduced a type of hyperring (
R
,
+
,
·
) where + is a hyperoperation and · is an ordinary binary operation. Such a hyperring is called a Krasner hyperring. Rota in [21] introduced a type of hyperring (
R
,
+
,
·
) where + is an ordinary binary operation and · is a hyperoperation. Such a hyperring is called a multiplicative hyperring. de Salvo [22] introduced and studied a type of hyperring (
R
,
+
,
·
) where + and · are hyperoperations. The most comprehensive reference for hyperrings is Davvaz and Leoreanu-Fotea’s book [18]. Some other references are [23–31].In this paper, we present and study associate, hyperdomainlike, and presimplifiable hyperrings. The relationships between these new hyperrings are presented.


## 2. Preliminaries

In this section, we will provide some definitions that will be used in the sequel. For full details about associate, domainlike, and presimplifiable rings, the reader should see [1, 4–6, 12]. Also, for details about hyperstructures and hyperrings, the reader should see [12].Definition 1.
LetR be a commutative ring with unity. (1)
R is called anassociate ring if whenever any two elements x
,
y
∈
R generate the same principal ideal of R, there is a unit u
∈
U
(
R
) such that x
=
u
y.
(2)
R is called adomainlike ring if all zero divisors of R are nilpotent.
(3)
R is called apresimplifiable ring if, for any two elements x
,
y
∈
R with x
=
x
y, we have x
=
0 or y
∈
U
(
R
).
(4)
R is called asuperassociate ring if every subring of R is associate.
(5)
R is called asuperpresimplifiable ring if every subring of R is presimplifiable.Example 2.
Integral domains, domainlike, and local rings are presimplifiable rings and hence are associate rings.Example 3 (see [6]).
It is easy to check thatZ
n is presimplifiable if and only if n
=
p
m, where p is some prime. Hence, Z
n is presimplifiable if and only if Z
n is local. Also, if a ring R is quasilocal, then R is presimplifiable, since J
(
R
)
=
M, the unique maximal ideal of R, and thus Z
(
R
)
⊆
M
=
J
(
R
).Definition 4.
LetH be a nonempty set and let ∘
:
H
×
H
→
P
*
(
H
) be ahyperoperation, where P
*
(
H
) is the family of all nonempty subsets of H. The couple (
H
,
∘
) is called ahypergroupoid. For any two nonempty subsets A and B of H and x
∈
H, one defines
(1)
A
∘
B
=
⋃
a
∈
A
,
b
∈
B
a
∘
b
,
A
∘
x
=
A
∘
{
x
}
,
x
∘
B
=
{
x
}
∘
B
.Definition 5.
A hypergroupoid(
H
,
∘
) is called asemihypergroup if for all a
,
b
,
c of H one has (
a
∘
b
)
∘
c
=
a
∘
(
b
∘
c
), which means that
(2)
⋃
u
∈
a
∘
b
u
∘
c
=
⋃
v
∈
b
∘
c
a
∘
v
.
A hypergroupoid (
H
,
∘
) is called aquasihypergroup if for all a of H one has a
∘
H
=
H
∘
a
=
H. This condition is also called thereproduction axiom.
A hypergroupoid(
H
,
∘
) which is both a semihypergroup and a quasihypergroup is called ahypergroup.Definition 6.
LetH be a nonempty set and let + be a hyperoperation on H. The couple (
H
,
+
) is called acanonical hypergroup if the following conditions hold: (1)
x
+
y
=
y
+
x, for all x
,
y
∈
H;
(2)
x
+
(
y
+
z
)
=
(
x
+
y
)
+
z, for all x
,
y
,
z
∈
H;
(3)
there exists a neutral element0
∈
H such that x
+
0
=
{
x
}
=
0
+
x, for all x
∈
H;
(4)
for everyx
∈
H, there exists a unique element -
x
∈
H such that 0
∈
x
+
(
-
x
)
∩
(
-
x
)
+
x;
(5)
z
∈
x
+
y implies y
∈
-
x
+
z and x
∈
z
-
y, for all x
,
y
,
z
∈
H.Definition 7.
LetR be a nonempty set and let + and · be hyperoperation and usual operation on R, respectively. The triple (
R
,
+
,
·
) is called aKrasner hyperring if the following axioms hold: (1)
(
R
,
+
) is a canonical hypergroup;
(2)
(
R
,
·
) is a semigroup having zero as a bilaterally absorbing element; that is, x
·
0
=
0
·
x
=
0 for all x
∈
R;
(3)
multiplication· is distributive over the hyperoperation +; that is, x
·
(
y
+
z
)
=
x
·
y
+
x
·
z and (
y
+
z
)
·
x
=
y
·
x
+
z
·
x for all x
,
y
,
z
∈
R.
If(
R
,
+
,
·
) is a hyperring, A and B are nonempty subsets of R and x
∈
R; then we define
(3)
A
+
B
=
{
y
:
y
∈
a
+
b
for some
a
∈
A
,
b
∈
B
}
,
A
·
B
=
{
y
:
y
∈
∑
i
=
1
n
a
i
b
i
,
a
i
∈
A
,
b
i
∈
B
,
n
∈
Z
+
}
,
A
+
x
=
A
+
{
x
}
,
x
+
B
=
{
x
}
+
B
,
-
A
=
{
-
a
:
a
∈
A
}
.
A hyperring(
R
,
+
,
·
) is called acommutative hyperring with unity if (
R
,
·
) is a commutative semigroup with unity.For anyx
,
y
∈
R, we use x
y instead of x
·
y.


## 3. Associate, Domainlike, and Presimplifiable Hyperrings

Throughout this section, all hyperrings will be assumed to be commutative Krasner hyperrings with unity.J
(
R
), N
(
R
), U
(
R
), and Z
(
R
) will denote Jacobson radical, nilradical, the set of units, and the set of zero divisors of R, respectively. Ann
(
x
) will denote the annihilator of 0
≠
x
∈
R.LetX be a subset of a hyperring R. Let {
A
i
∣
i
∈
J
} be the family of all hyperideals in R which contain X. Then, ⋂
i
∈
J
A
i is called thehyperideal generated by
X. This hyperideal is denoted by 〈
X
〉. If X
=
{
x
1
,
…
,
x
n
}, then the hyperideal 〈
X
〉 is denoted by 〈
x
1
,
…
,
x
n
〉.Lemma 8 (see [18]).
LetR be a hyperring and a
∈
R. Then, (1)
R
a
=
a
R is a hyperideal of R;
(2)
ifR has a unit element, then R
a
=
〈
a
〉
=
a
R.Definition 9.
LetR be a hyperring and let ρ
~, ρ
≈, and ρ
≅ be relations on R defined as follows. (1)
x
ρ
~
y if and only if 〈
x
〉
=
〈
y
〉, for all x
,
y
∈
R.
(2)
x
ρ
≈
y if and only if x
=
u
y, for all x
,
y
∈
R and some u
∈
U
(
R
).
(3)
x
ρ
≅
y if and only if x
ρ
~
y and when x
≠
0 and x
=
r
y, then r
∈
U
(
R
).Theorem 10.
LetR be a hyperring. Then, (1)
ρ
~ is an equivalence relation on R,
(2)
ρ
≈ is an equivalence relation on R,
(3)
ρ
≅ is an equivalence relation on R if and only if, for all x
,
y
∈
R, x
=
x
y implies x
=
0 or y
∈
U
(
R
).Proof.
(1) and (2) are clear. For (3), suppose thatρ
≅ is an equivalence on R and suppose that x
=
x
y with x
≠
0. Then, x
a
=
0 for some a
∈
1
-
y which implies y
∈
1
-
a. But then, a
=
0 so that y
∈
U
(
R
). The converse is straightforward.Definition 11.
IfR is a hyperring, then a
≠
0 in R is said to be azero divisor if there exists b
≠
0 in R such that a
b
=
0.Definition 12.
A hyperring is ahyperdomain if it has no zero devisor.The following example is presented in [18, 29, 30].Example 13.
Let(
G
,
·
) be a finite group with m elements, m
>
3, and define a hyperaddition and a multiplication on H
=
G
∪
{
0
}, by
(4)
a
+
0
=
0
+
a
=
{
a
}
,
∀
a
∈
H
,
a
+
a
=
{
a
,
0
}
,
∀
a
∈
G
,
a
+
b
=
b
+
a
=
H
∖
{
a
,
b
}
,
∀
a
,
b
∈
G
,
a
≠
b
,
a
⊙
0
=
0
,
∀
a
∈
H
,
a
⊙
b
=
a
·
b
,
∀
a
,
b
∈
G
.
Then, (
H
,
+
,
⊙
) is a hyperring.Example 14.
It is easy to see that Example13 is a hyperdomain.Theorem 15.
LetR be a hyperring. (1)
x
ρ
≅
y implies x
ρ
~
y, for all x
,
y
∈
R.
(2)
x
ρ
≈
y implies x
ρ
~
y, for all x
,
y
∈
R.
(3)
IfR is hyperdomain, then x
ρ
~
y implies x
ρ
≈
y, for all x
,
y
∈
R.Proof.
(1) and (2) are obvious. For (3), suppose thatR is hyperdomain and suppose that x
ρ
~
y. Then, 〈
x
〉
=
〈
y
〉 which implies that x
=
k
1
y and y
=
k
2
x for some k
1
,
k
2
∈
R. Thus, x
=
k
1
k
2
x which implies x
a
=
0 for some a
∈
1
-
k
1
k
2
⊆
R so that k
1
k
2
∈
1
-
a. Since R is a hyperdomain, if x
≠
0, then a
=
0, and, therefore, k
1
k
2
∈
U
(
R
) which implies k
1
,
k
2
∈
U
(
R
). Hence, x
ρ
≈
y.Definition 16.
LetR be a hyperring. (1)
R is said to be anassociate hyperring if x
ρ
~
y implies x
ρ
≈
y for all x
,
y
∈
R.
(2)
R is said to be asuperassociate hyperring if every subhyperring of R is associate.Example 17.
The hyperring in Example13 is associate hyperring and superassociate hyperring too.Definition 18.
LetR be a hyperring. (1)
R is said to be apresimplifiable hyperring if, for all x
,
y
∈
R, x
=
x
y implies x
=
0 or y
∈
U
(
R
).
(2)
R is said to be asuperpresimplifiable hyperring if every subhyperring of R is presimplifiable.
(3)
R is said to be ahyperdomainlike hyperring if all zero divisors of R are nilpotent.Example 19.
LetR
1
=
Z
p
m, R
2
=
Z
p
n, and R
3
=
Z
p
k be three rings, where m
,
n
>
k and p is a prime number. Suppose that M
i, i
=
1,2
,
3, are maximal ideals of R
i, i
=
1,2
,
3. Let H
i, i
=
1,2
,
3, be subsets of R
i and define a hyperaddition and multiplication on H
i by
(5)
x
⊕
i
y
=
x
+
y
+
M
i
,
i
=
1,2
,
3
,
∀
x
,
y
∈
H
i
,
x
⊙
y
=
x
y
,
∀
x
,
y
∈
H
i
.
Then, (
H
i
,
⊕
i
,
⊙
) are hyperrings. Since R
i, i
=
1,2
,
3, are domainlike (see Example 3), consequently each H
i is hyperdomainlike.Theorem 20.
LetR be a hyperring. The following conditions are equivalent: (1)
x
ρ
~
y implies x
ρ
≅
y, for all x
,
y
∈
R;
(2)
x
ρ
≈
y implies x
ρ
≅
y, for all x
,
y
∈
R;
(3)
x
ρ
≅
x for all x
∈
R;
(4)
R is presimplifiable;
(5)
Z
(
R
)
⊆
1
-
U
(
R
);
(6)
Z
(
R
)
⊆
J
(
R
);
(7)
for0
≠
x
∈
R, y
〈
x
〉
=
〈
x
〉 implies y
∈
U
(
R
).Proof.
(1
⇒
2): it follows from the definitions of ρ
~
,
ρ
≈, and ρ
≅.
(2
⇒
3): suppose that x
ρ
≈
y which implies that x
ρ
≅
y for all x
,
y
∈
R. Then, x
ρ
≈
x implies x
=
1
·
x and, therefore, x
ρ
≅
x for all x
∈
R.
(3
⇒
4): suppose that x
ρ
≅
x for all x
∈
R and suppose that x
=
x
y. Then, x
a
=
0 for some a
∈
1
-
y which implies y
∈
1
-
a
⊆
R. If x
≠
0, we must have y
∈
U
(
R
). Hence, R is presimplifiable.
(4
⇒
5): suppose that R is presimplifiable and suppose that x
∈
Z
(
R
). Then, there exists 0
≠
y
∈
R such that x
y
=
0. But then, y
∈
y
-
x
y which implies that y
=
y
a for some a
∈
1
-
x
⊆
U
(
R
) so that x
∈
1
-
U
(
R
).
(5
⇒
6): suppose that x
∈
Z
(
R
). Then, y
=
x
r
∈
Z
(
R
) for all 0
≠
r
∈
R. Thus, y
∈
1
-
u for some u
∈
U
(
R
). Now, u
∈
1
-
y implies 1
-
y
⊆
U
(
R
) and, therefore, x
∈
J
(
R
).
(6
⇒
7): suppose that 0
≠
x
∈
R and y
〈
x
〉
=
〈
x
〉. Then, x
=
y
x
r for some r
∈
R so that x
a
=
0 for some a
∈
1
-
y
r. Since x
≠
0, we must have a
∈
Z
(
R
)
⊆
J
(
R
) so that y
r
∈
1
-
a
⊆
U
(
R
) and, therefore, y
∈
U
(
R
).
(7
⇒
1): suppose that x
ρ
~
y and x
≠
0. Then, 〈
x
〉
=
〈
y
〉 and x
=
r
y for some r
∈
R. Thus, x
〈
x
〉
=
r
y
〈
y
〉
=
r
〈
y
〉 and we have 〈
x
〉
=
r
〈
x
〉 which implies r
∈
U
(
R
). Hence, x
ρ
≅
y.Theorem 21.
LetR be a hyperring. If R is quasilocal, then it is presimplifiable.Proof.
Suppose thatR is quasilocal. Then, R has a unique maximal hyperideal M and J
(
R
)
=
M. Hence, Z
(
R
)
⊆
M
=
J
(
R
) and therefore R is presimplifiable.Theorem 22.
LetR be a hyperring. If R is presimplifiable, then R is associate.Proof.
It follows from Theorem20.Corollary 23.
Any hyperdomain or any quasilocal hyperring is an associate hyperring.Proof.
This is immediate from Theorems20, 21, and 22.Theorem 24.
LetR be a hyperring. Then, the following statements are equivalent: (1)
R is hyperdomainlike;
(2)
Z
(
R
)
⊆
N
(
R
);
(3)
〈
0
〉 is a primary hyperideal of R.Proof.
(1
⇒
2): it is obvious.
(2
⇒
3): let x
y
∈
〈
0
〉. Then, x
y
=
0. If y
≠
0, then x
∈
Z
(
R
)
⊆
N
(
R
) so that x
n
=
0 for some positive integer n and, therefore, x
n
∈
〈
0
〉. Hence, 〈
0
〉 is primary.
(3) implies (1): suppose that〈
0
〉 is a primary hyperideal of R. Let x
∈
Z
(
R
) be arbitrary. Then, there exists 0
≠
y
∈
R such that x
y
=
0 which implies that x
y
∈
〈
0
〉. Since y
∉
〈
0
〉, we must have x
n
∈
〈
0
〉 and therefore x
n
=
0. Hence, x is nilpotent. Since x is arbitrary, it follows that R is hyperdomainlike.LetR
1 and R
2 be any two hyperrings and let R
=
R
1
×
R
2. For any (
a
,
b
)
,
(
c
,
d
)
∈
R, define
(6)
(
a
,
b
)
+
(
c
,
d
)
=
{
(
x
,
y
)
:
x
∈
a
+
c
,
y
∈
b
+
d
}
,
(
a
,
b
)
·
(
c
,
d
)
=
(
a
·
c
,
b
·
d
)
.
Then, (
R
,
+
,
·
) is a hyperring called thedirect product of R
1 and R
2.Theorem 25.
LetR be a hyperring. (1)
IfR is hyperdomainlike, then it is superpresimplifiable.
(2)
IfR is presimplifiable (associate), then a subhyperring of R need not be presimplifiable (associate).
(3)
IfR is superassociate, then it need not be presimplifiable.
(4)
A direct product of superassociate hyperrings need not be superassociate.Proof.
See Anderson et al. [4].Theorem 26.
LetR
1 and R
2 be any two presimplifiable hyperrings with the set of units {
±
1
}. Then, R
=
R
1
×
R
2 is a superassociate hyperring.Proof.
The proof is similar to the proof of Theorem  2.1 in [12].Theorem 27.
LetA be a hyperideal of a hyperring R. A is primary if and only if R
/
A is hyperdomainlike.Proof.
Suppose thatA is primary. Let x
¯
∈
Z
(
R
/
A
). Then, there exists 0
¯
≠
y
¯
∈
R
/
A such that x
¯
y
¯
=
0
¯ which implies that x
y
+
A
⊆
A which implies x
y
∈
A. Since A is primary and y
∉
A, it follows that x
n
∈
A from which we have x
¯
n
=
0
¯ and therefore x
¯
∈
N
(
R
/
A
). Hence, Z
(
R
/
A
)
⊆
N
(
R
/
A
) and, thus, R
/
A is hyperdomainlike.
Conversely, suppose thatR
/
A is hyperdomainlike. Since A is a zero of R
/
A, it follows that A is primary.Theorem 28.
LetR be a hyperring. If R is hyperdomainlike, then R is presimplifiable.Proof.
The proof follows from Theorem21, since Z
(
R
)
⊆
N
(
R
)
⊆
J
(
R
).Theorem 29.
LetR be a hyperring. If R is hyperdomainlike, then R
/
N
(
R
) is a hyperdomain.Proof.
Suppose thatR is hyperdomainlike. Let x
¯
,
y
¯
∈
R
/
N
(
R
) such that x
¯
y
¯
=
0
¯ with 0
¯
≠
x
¯. Then, x
y
+
N
(
R
)
⊆
N
(
R
) implies x
y
∈
N
(
R
) but x
∉
N
(
R
). For some positive integer n, we have (
x
y
)
n
=
0 which implies x
n
y
n
∈
〈
0
〉. Since x
n
∉
〈
0
〉, we must have (
y
n
)
k
∈
〈
0
〉 for some positive integer k. Hence, y
∈
N
(
R
) and, therefore, y
¯
=
0
¯. Hence, R
/
N
(
R
) is a hyperdomain.Theorem 30.
LetR be a hyperring. If R is hyperdomainlike, then Z
(
R
) is the unique minimal prime hyperideal of R.Proof.
Suppose thatR is hyperdomainlike. Then, Z
(
R
)
⊆
N
(
R
). Since N
(
R
) is the intersection of all prime hyperideals of R, the required result follows.Theorem 31.
LetR be a hyperring and let A be a hyperideal of R. Then, R
/
r
(
A
) is hyperdomainlike if and only if R
/
r
(
A
) is a hyperdomain.Proof.
Suppose thatR
/
r
(
A
) is a hyperdomain. Then, r
(
A
) is a prime hyperideal of R which is clearly a primary hyperideal of R
/
r
(
A
). Hence, R
/
r
(
A
) is hyperdomainlike.
Conversely, suppose thatR
/
r
(
A
) is hyperdomainlike. Let x
¯
,
y
¯
∈
R
/
r
(
A
) such that x
¯
y
¯
=
0
¯ with 0
¯
≠
x
¯. Then, x
y
+
r
(
A
)
⊆
r
(
A
) implies x
y
∈
r
(
A
) but x
∉
r
(
a
). Since r
(
A
) is primary in R
/
r
(
A
), we must have y
n
∈
r
(
A
) for some positive integer n. Thus, (
y
n
)
k
∈
A for some positive integer k. Thus, y
∈
r
(
A
) and, therefore, y
¯
=
0
¯. Hence, R
/
r
(
A
) is a hyperdomain.Theorem 32.
LetR be a hyperring. R
/
N
(
R
) is a hyperdomainlike ring if and only R
/
N
(
R
) is a hyperdomain.Proof.
The proof is similar to the proof of Theorem29.Theorem 33.
LetR be a hyperring and let S
-
1
R be the total quotient hyperring of R. If R is hyperdomainlike so also is S
-
1
R.Proof.
The proof is similar to the classical ring and therefore omitted.Theorem 34.
LetR be a hyperring. R
/
N
(
R
) is presimplifiable if and only if whenever x
=
x
y and x
∉
N
(
R
), then y
∈
U
(
R
).Proof.
Suppose thatR
/
N
(
R
) is presimplifiable and suppose that x
=
x
y with x
∉
N
(
R
). Then, x
¯
=
x
¯
y
¯ and x
¯
≠
0
¯ so that y
¯
∈
U
(
R
/
N
(
R
)
). Now, x
-
x
y
⊆
N
(
R
) which implies that x
a
∈
N
(
R
) for some a
∈
1
-
y and therefore x
n
a
n
=
0 and a
n
∈
1
-
y
b, where b
∈
∑
r
=
1
n
(
-
1
)
r
-
1
(
n
r
)
y
r
-
1 and y
b
∈
1
-
a
n so that x
n
y
b
∈
x
n
-
x
n
a
n. Since x
n
≠
0, we must have y
b
∈
U
(
R
) and hence y
∈
U
(
R
).
Conversely, suppose thatx
=
x
y with x
∉
N
(
R
) implies that y
∈
U
(
R
). Then, x
¯
=
x
¯
y
¯ and x
¯
≠
0
¯. It can be shown that y
¯
∈
U
(
R
/
N
(
R
)
). Hence, R
/
N
(
R
) is presimplifiable.Corollary 35.
LetR be a hyperring. If R is presimplifiable, then R
/
N
(
R
) is presimplifiable and hence strongly associate.Definition 36.
LetA be a hyperideal of a hyperring R and let A
*
=
A
-
{
0
}. A is called apresimplifiable hyperideal if, for every x
∈
A
* and y
∈
R, x
=
x
y implies that y
∈
U
(
R
).Lemma 37.
Every hyperideal of a presimplifiable hyperringR is presimplifiable.Proof.
It is obvious.Theorem 38.
LetA be a hyperideal of a hyperring R. Then, A is presimplifiable if and only if A
n
n
(
x
)
⊆
J
(
R
) for all x
∈
A
*.Proof.
Suppose thatA is presimplifiable. Let y
∈
Ann
(
x
), where x
∈
A
*. Then, x
y
=
0 and we can write x
=
x
a for some a
∈
1
-
r
y and r
∈
R. Since R is presimplifiable, we have a
∈
U
(
R
) so that 1
-
r
y
⊆
U
(
R
). Hence, y
∈
J
(
R
).
Conversely, suppose thatAnn
(
x
)
⊆
J
(
R
) for all x
∈
A
*. Let x
=
x
y. Then, x
a
=
0 for some a
∈
1
-
y. Thus, a
∈
Ann
(
x
)
⊆
J
(
R
) and, therefore, 1
-
y
∈
J
(
R
) so that y
∈
U
(
R
). Hence, A is presimplifiable.Theorem 39.
LetR be a hyperring. Then, R is presimplifiable if and only if R
/
N
(
R
) and N
(
R
) are presimplifiable.Proof.
Suppose thatR is presimplifiable. Let x
¯
∈
Z
(
R
/
N
(
R
)
)
-
0
¯. Then, there exits 0
¯
≠
y
∈
Z
(
R
/
N
(
R
)
)
-
{
0
¯
} such that x
¯
y
¯
=
0
¯ so that x
y
+
N
(
R
)
⊆
N
(
R
) which implies that x
y
∈
N
(
R
) and, for a positive integer n, we have x
n
y
n
=
0. Since y
n
≠
0, we must have x
n
∈
Z
(
R
) and so x
∈
Z
(
R
)
⊆
J
(
R
). Hence, x
¯
∈
J
(
R
/
N
(
R
)
) and, therefore, R
/
N
(
R
) is presimplifiable.
Conversely, suppose thatR
/
N
(
R
) and N
(
R
) are presimplifiable. Let 0
≠
x
,
y
∈
R such that x
=
x
y. Then, x
¯
=
x
¯
y
¯. Obviously, x
∈
N
(
R
) and y
¯
∈
U
(
R
/
N
(
R
)
). Consequently, y
∈
U
(
R
) and, therefore, R is presimplifiable.Definition 40.
LetR
1 and R
2 be any two hyperrings and let ϕ
:
R
1
→
R
2 be a mapping from R
1 into R
2. (1)
ϕ is called ahomomorphism if
(i)
ϕ
(
a
+
b
)
⊆
ϕ
(
a
)
+
ϕ
(
b
), for all a
,
b
∈
R
1,
(ii)
ϕ
(
a
b
)
=
ϕ
(
a
)
ϕ
(
b
), for all a
,
b
∈
R
1,
(iii)
ϕ
(
0
)
=
0.
(2)
ϕ is called agood orstrong homomorphism if
(i)
ϕ
(
a
+
b
)
=
ϕ
(
a
)
+
ϕ
(
b
), for all a
,
b
∈
R
1,
(ii)
ϕ
(
a
b
)
=
ϕ
(
a
)
ϕ
(
b
), for all a
,
b
∈
R
1,
(iii)
ϕ
(
0
)
=
0.
(3)
A strong homomorphismϕ from a hyperring R
1 into a hyperring R
2 is called anisomorphism if ϕ is bijective and we write R
1
≅
R
2.
(4)
Ifϕ is a homomorphism from a hyperring R
1 into a hyperring R
2, then thekernel of ϕ denoted by Ker
⁡
ϕ is the set {
x
∈
R
1
:
ϕ
(
x
)
=
0
} and the image of ϕ denoted by Im
⁡
ϕ is the set {
ϕ
(
x
)
:
x
∈
R
1
}. It is known that Ker
⁡
ϕ is a hyperideal of R
1 and Im
⁡
ϕ is a hyperideal of R
2.Definition 41.
LetR
1, R
2, and R
3 be any three hyperrings with strong homomorphisms ϕ
i
:
R
i
→
R
3, i
=
1,2, which preserve the unity. The set P
=
{
(
r
1
,
r
2
)
∈
R
1
×
R
2
:
ϕ
1
(
r
1
)
=
ϕ
2
(
r
2
)
} is called thepullback of R
1
→
R
3
←
R
2.Lemma 42.
LetR
1, R
2, and R
3 be any three hyperrings and let P be the pullback of R
1
→
R
3
←
R
2. Then, P is a subhyperring of R
1
×
R
2 and U
(
P
), the set of units of P, is {
(
u
1
,
u
2
)
∈
P
:
u
1
∈
U
(
R
1
)
,
u
2
∈
U
(
R
2
)
}.Theorem 43.
LetR
1, R
2, and R
3 be any three hyperrings with strong homomorphisms ϕ
i
:
R
i
→
R
3, i
=
1,2. If Ker
⁡
ϕ
1
⊈
Z
(
R
1
) and there exists u
∈
U
(
R
1
) with ϕ
1
(
u
)
,
ϕ
1
(
u
-
1
)
∈
ϕ
2
(
R
2
)
∖
ϕ
2
(
U
(
R
2
)
), then the pullback P of R
1
→
R
3
←
R
2 is not an associate hyperring.Proof.
The proof is similar to the proof of Proposition  6 in [4].Theorem 44.
LetR
1, R
2, and R
3 be any three hyperrings with strong epimorphisms ϕ
i
:
R
i
→
R
3, i
=
1,2, which are not strong isomorphisms. If R
1 and R
2 are hyperdomains, then the pullback P of R
1
→
R
3
←
R
2 is associate (presimplifiable) if and only if ϕ
1
(
U
(
R
1
)
)
=
ϕ
2
(
U
(
R
2
)
), respectively, ϕ
i
-
1
(
1
)
⊆
U
(
R
i
), i
=
1,2, or, equivalently, ϕ
i
-
1
(
U
(
R
3
)
=
U
(
R
i
), i
=
1,2.Proof.
The proof is similar to the proof of Theorem  7 in [4].Theorem 45.
LetR
1, R
2, and R
3 be any three hyperrings with strong epimorphisms ϕ
i
:
R
i
→
R
3, i
=
1,2, which are not strong isomorphisms. If R
1 and R
2 are presimplifiable, then the pullback P of R
1
→
R
3
←
R
2 is presimplifiable if and only if ϕ
i
-
1
(
1
)
⊆
U
(
R
i
), i
=
1,2.Proof.
The proof is similar to the proof of Theorem  2.3 in [12].Theorem 46.
LetR
1, R
2, and R
3 be any three hyperrings with strong epimorphisms ϕ
i
:
R
i
→
R
3, i
=
1,2, which are not strong isomorphisms. If R
1 and R
2 are hyperdomainlikes, then the pullback P of R
1
→
R
3
←
R
2 is hyperdomainlike if and only if Ker
⁡
ϕ
i
⊆
Z
(
R
i
), i
=
1,2.Proof.
The proof is similar to the proof of Theorem  2.5 in [12].Corollary 47.
LetR
1, R
2, and R
3 be any three hyperrings with strong epimorphisms ϕ
i
:
R
i
→
R
3, i
=
1,2, which are not strong isomorphisms. If R
1 and R
2 are hyperdomainlikes, then the pullback P of R
1
→
R
3
←
R
2 is associate but not hyperdomainlike if and only if Ker
⁡
ϕ
i
⊈
Z
(
R
i
), i
=
1,2, and ϕ
1
(
U
(
R
1
)
)
=
ϕ
2
(
U
(
R
2
)
).



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*Source: 458603-2014-08-14.xml*