Derivations of gamma (semi)hyperrings
Differential $\Gamma$ -(semi)hyperrings are $\Gamma$ -(semi)hyperrings equipped with derivation, which is a linear unary function satisfying the Leibniz product rule.We introduce the notions of derivation and weak derivation on $\Gamma$ -hyperrings and $\Gamma$ -semihyperrings and obtain some import...
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| author | Ardekani, L. K. Davvaz, B. Ардекані, Л. К. Давваз, Б. |
| author_facet | Ardekani, L. K. Davvaz, B. Ардекані, Л. К. Давваз, Б. |
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| description | Differential $\Gamma$ -(semi)hyperrings are $\Gamma$ -(semi)hyperrings equipped with derivation, which is a linear unary function satisfying
the Leibniz product rule.We introduce the notions of derivation and weak derivation on $\Gamma$ -hyperrings and $\Gamma$ -semihyperrings
and obtain some important results relating to them in a specific way. |
| first_indexed | 2026-03-24T02:09:07Z |
| format | Article |
| fulltext |
UDC 512.5
L. K. Ardekani, B. Davvaz (Yazd Univ., Iran)
DERIVATIONS OF GAMMA (SEMI)HYPERRINGS
ПОХIДНI ГАММА (НАПIВ)ГIПЕРКIЛЕЦЬ
Differential \Gamma -(semi)hyperrings are \Gamma -(semi)hyperrings equipped with derivation, which is a linear unary function satisfying
the Leibniz product rule. We introduce the notions of derivation and weak derivation on \Gamma -hyperrings and \Gamma -semihyperrings
and obtain some important results relating to them in a specific way.
Диференцiальнi \Gamma -(напiв)гiперкiльця — це \Gamma -(напiв)гiперкiльця з похiдною, що є лiнiйною унарною функцiєю i
задовольняє правило добутку Лейбнiца. Введено поняття похiдної та слабкої похiдної на \Gamma -гiперкiльцях та \Gamma -
напiвгiперкiльцях та отримано деякi важливi результати, щo стосуються цих понять у конкретному виглядi.
1. Introduction. The notion of \Gamma -ring was introduced by N. Nobusawa in 1964 [19]. After him
in 1966, Barnes extended this notion and obtained more results [2]. Almost ten years later Kyuno
in [15, 16] investigated new aspects of \Gamma -rings. The notion of \Gamma -semirings was introduced by
Rao [21] as a generalization of \Gamma -ring as well as of semiring. In fact, every semiring (R,+, \circ ) can
be considered as a \Gamma -semiring by putting \Gamma = \{ \circ \} . In this way, many classical notions of semiring
have been extended to \Gamma -semiring.
The theory of algebraic hyperstructures is a well established branch of classical algebraic theory.
In 1934, at the 8th Congress of Scandinavian Mathematicians, Marty [18] has introduced, for the first
time, the notion of hypergroup, using it in different contexts: algebraic functions, rational fractions
and noncommutative groups. 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. One of
the first books, dedicated especially to hypergroups, is written by Corsini in 1993 [3]. Another book
on this subject was published one year later [24]. A recent book on hyperstructures [4] points out on
their applications in fuzzy and rough set theory, cryptography, codes, automata, probability, geometry,
lattices, binary relations, graphs and hypergraphs. Another book is devoted especially to the study
of hyperring theory, written by Davvaz and Leoreanu-Fotea [5]. It begins with some basic results
cocerning ring theory and algebraic hyperstructures, which represent the most general algebraic
context, in which the reality can be modelled. After these subjects, several kinds of hyperrings are
introduced and analyzed. The volume ends with an outline of applications in chemistry and physics,
analyzing several special kinds of hyperstructures: e-hyperstructures and transposition hypergroups.
Now, we recall some preliminary definitions from [5]. For more details and properties, we refer the
readers to [3, 5, 24].
Let H be a nonempty set and \circ : H \times H \rightarrow P \ast (H) be a hyperoperation, where P \ast (H) is a
family of all nonempty subsets of H. The couple (H, \circ ) is called a hypergroupoid. For any two
nonempty subsets A and B of H and x \in H, we define A \circ B =
\bigcup
a\in A,b\in B a \circ b, A \circ \{ x\} = A \circ x
and \{ x\} \circ A = x \circ A. A hypergroupoid (H, \circ ) is called a semihypergroup if (x \circ y) \circ z = x \circ (y \circ z),
for all x, y, z \in H. In addition, if for every x \in H, x \circ H = H = H \circ x, then (H, \circ ) is called
c\bigcirc L. K. ARDEKANI, B. DAVVAZ, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8 1011
1012 L. K. ARDEKANI, B. DAVVAZ
a hypergroup. A nonempty subset K of a semihypergroup (H, \circ ) is called a subsemihypergroup
if it is a semihypergroup. In other words, a nonempty subset K of a semihypergroup (H, \circ ) is
a subsemihypergroup if K \circ K \subseteq K. We say that a hypergroup (H, \circ ) is canonical if (1) it is
commutative; (2) it has a scalar identity (also called scalar unit), which means that there exists
e \in H such that for every x \in H, we have x \circ e = \{ x\} ; (3) every element has a unique inverse,
which means that for all x \in H, there exists a unique x - 1 \in H such that e \in x - 1 \circ x; (4) it is
reversible, which means that if x \in y \circ z, then z \in y - 1 \circ x and y \in x \circ z - 1.
As a generalization of semihypergroups, Davvaz et al. studied the notion of \Gamma -semihypergroups
in [1, 12]: Let S and \Gamma be two nonempty sets. Then S is called a \Gamma -semihypergroup if x\alpha y \subseteq S
and x\alpha (y\beta z) = (x\alpha y)\beta z, for all \alpha , \beta \in \Gamma and x, y, z \in S. \Gamma -semihypergroup S is called a
\Gamma -hypergroup if (S, \gamma ) is a hypergroup, for all \gamma \in \Gamma .
The more general structure that satisfies the ring-like axioms is the hyperring in the general sense:
(R,+, \cdot ) is a hyperring if “+” and “\cdot ” are two hyperoperations such that (R,+) is a hypergroup and
“\cdot ” is an associative hyperoperation which is distributive with respect to “+” . There are different
notions of hyperrings. If only the addition “+” is a hyperoperation and the multiplication “\cdot ” is a
usual operation, then R is called Krasner hyperring. Exactly, a Krasner hyperring is an algebraic
structure (R,+, \cdot ) which satisfies the following axioms: (1) (R,+) is a canonical hypergroup (the
scaler identity of (R,+) is denoted by 0); (2) (R, \cdot ) is a semigroup having zero as a bilaterally
absorbing element, i.e., x \cdot 0 = x; (3) the multiplication is distributive with respect to the hyper-
operation +. In [9], \Gamma -hyperrings as a generalization of hyperrings are introduced: Let (R,+) be
a canonical hypergroup and \Gamma be a nonempty set. Then R is called a \Gamma -hyperring if there exists
a mapping R \times \Gamma \times R - \rightarrow P \ast (R) (images denoted by x\alpha y, for all x, y \in R and \alpha \in \Gamma ), where
P \ast (R) is the set of all nonempty subsets of R, satisfying the following conditions: (1) there exists a
zero element that a bilaterally absorbing element, i.e., x\alpha 0 = 0\alpha x = 0 and x+0 = x, for all \alpha \in \Gamma
and x \in R; (2) x\alpha (y + z) = x\alpha y + x\alpha z; (3) (x + y)\alpha z = x\alpha z + y\alpha z; (4) x\alpha (y\beta z) = (x\alpha y)\beta z.
Let (R,+) be a commutative semihypergroup and (\Gamma ,+) be a commutative group. Then R is called
a \Gamma -semihyperring if there exists a mapping R \times \Gamma \times R - \rightarrow P \ast (R) (images denoted by x\alpha y, for
all x, y \in R and \alpha \in \Gamma ) satisfying the conditions (2), (3), (4) and (5) x(\alpha + \beta )y = x\alpha y + x\beta y
[8]. In the above definition if R is a semigroup, then R is called a multiplicative \Gamma -semihyperring
[8]. We recall the following definitions and notions of [9]: A \Gamma -semihyperring (\Gamma -hyperring) R
is called commutative (weak commutative, respectively) if x\alpha y = y\alpha x (x\alpha y \cap y\alpha x \not = \varnothing , respec-
tively), for all x, y \in R and \alpha \in \Gamma . The meaning of center (weak center, respectively) of R is
Z(R) = \{ x \in R | x\alpha y = y\alpha x, for all y \in R and \alpha \in \Gamma \}
\bigl(
Z\circ (R) =
\bigl\{
x \in R | x\alpha y \cap y\alpha x \not = \varnothing , for
all y \in R and \alpha \in \Gamma
\bigr\}
, respectively
\bigr)
. We say that R is a \Gamma -semihyperring with zero, if there exists
0 \in R such that x \in x+ 0, 0 \in 0\alpha x and 0 \in x\alpha 0, for all x \in R and \alpha \in \Gamma . Let A and B be two
nonempty subsets of \Gamma -semihyperring (\Gamma -hyperring) R. We define
A+B = \{ t \in R | t \in a+ b, a \in A, b \in B\} ,
A\Gamma B = \{ t \in R | t \in a\alpha b, a \in A, b \in B,\alpha \in \Gamma \} ,
A\Gamma
\sum
B =
\Bigl\{
t \in R | t \in
\sum n
i=1
ai\alpha ibi, ai \in A, bi \in B,\alpha i \in \Gamma , n \in \BbbN
\Bigr\}
,
\BbbN X =
\Bigl\{
t \in R | t \in
\sum n
i=1
nixi, xi \in X, n, ni \in \BbbN
\Bigr\}
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
DERIVATIONS OF GAMMA (SEMI)HYPERRINGS 1013
A nonempty subset S of \Gamma -semihyperring (\Gamma -hyperring, respectively) R is called a \Gamma -subsemihyper-
ring (\Gamma -subhyperring, respectively) if it is closed with respect to the hypermultiplication and hyper-
addition. In other words, a nonempty subset S of \Gamma -semihyperring R is a \Gamma -subsemihyperring if
S + S \subseteq S and S\Gamma S \subseteq S. A right ideal (left ideal, respectively) I of a \Gamma -semihyperring R is an
additive subsemihypergroup (R,+) such that I\Gamma R \subseteq I (R\Gamma I \subseteq I, respectively). If I is both right
and left ideal of R, then we say that I is a two-sided ideal or simply an ideal of R. An ideal of
\Gamma -hyperring R defines similarly. Let X be a nonempty subset of \Gamma -semihyperring (\Gamma -hyperring)
R. By the term left ideal
\bigl\langle
X
\bigr\rangle
l
(right ideal
\bigl\langle
X
\bigr\rangle
r
, respectively) of R generated by X, we mean the
intersection of all left ideals (right ideals, respectively) of R contains X. Hence,
(1)
\bigl\langle
X
\bigr\rangle
l
= \BbbN X +R\Gamma
\sum
X;
(2)
\bigl\langle
X
\bigr\rangle
r
= \BbbN X +X\Gamma
\sum
R;
(3)
\bigl\langle
X
\bigr\rangle
= \BbbN X +R\Gamma
\sum
X +X\Gamma
\sum
R+R\Gamma
\sum
X\Gamma
\sum
R.
2. Derivation on \bfGamma -(semi)hyperrings. The concept of derivation on rings has been introduced
by Posner [17]. In 1987, Jing introduced the notion of derivation on \Gamma -rings [14]. After him,
many mathematicians studied derivation on rings and \Gamma -rings (see, for example, [6, 13, 20, 22, 23]).
In this section, we introduce the notions of derivation and weak derivation on \Gamma -hyperrings and
\Gamma -semihyperrings. Also, some properties of them are explored.
Definition 2.1. Let (R,+,\Gamma ) be a \Gamma -semihyperring (\Gamma -hyperring). The function d : R - \rightarrow R
is called derivation if for all x, y \in R and \alpha \in \Gamma , (1) d(x + y) = d(x) + d(y); (2) d(x\alpha y) =
= d(x)\alpha y + x\alpha d(y). The function d : R \rightarrow R is called weak derivation if for all x, y \in R, it
satisfies in (1) and (3) d(x\alpha y) \subseteq d(x)\alpha y + x\alpha d(y).
It is clear that every derivation is a weak derivation. Let (R,+,\Gamma ) be a \Gamma -hyperring and d be a
derivation on R. Then by the second condition of definition of derivation we have
d(0) = d(0\alpha 0) = d(0)\alpha 0 + 0\alpha d(0) = 0, where \alpha \in \Gamma .
The set of all derivations (weak derivations, respectively) of \Gamma -semihyperring (\Gamma -hyperring) (R,+,\Gamma )
is denoted by \Delta (R,+,\Gamma ) (D(R,+,\Gamma ), respectively). Note that \Delta (R,+,\Gamma ) \subseteq D(R,+,\Gamma ) \subseteq
\subseteq \mathrm{H}\mathrm{o}\mathrm{m}(R,+). The following examples illustrate the above concepts.
Example 2.1. Let R be a \Gamma -semihyperring with zero. Then the function d : R - \rightarrow R defined
by d(x) = 0, for all x \in R, is a weak derivation. If R is a \Gamma -hyperring, then d is a derivation.
Example 2.2. Let (R,+, \circ ) be a semihyperring such that x \circ y = x \circ y + x \circ y and \Gamma be a
commutative group. Define x\alpha y = x \circ y, for all x, y \in R and \alpha \in \Gamma . Then R is a \Gamma -semihyperring
[8]. It is easily to check that the function d : R - \rightarrow R defined by d(x) = x, for all x \in R, is a
derivation on (R,+,\Gamma ).
Example 2.3. Let R = \BbbZ 4 and \Gamma = \{ 0, 2\} . Then R is a multiplicative \Gamma -semihyperring with
the hyperoperation x\alpha y =
\bigl\{
0, 2
\bigr\}
, for all x, y \in R and \alpha \in \Gamma [8]. There exist only two derivations
on R, they are as d1(x) = x, for all x \in R and
d2(x) =
\left\{
0 \mathrm{i}\mathrm{f} x = 0,
1 \mathrm{i}\mathrm{f} x = 3,
2 \mathrm{i}\mathrm{f} x = 2,
3 \mathrm{i}\mathrm{f} x = 1.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1014 L. K. ARDEKANI, B. DAVVAZ
Also, there exist only four weak derivations on R, they are as d1, d2, d3(x) = 0 and
d4(x) =
\left\{ 0 \mathrm{i}\mathrm{f} x = 0,2,
2 \mathrm{i}\mathrm{f} x = 1,3.
Consequently, \Delta (\BbbZ 4,+4,\Gamma ) = \{ d1, d2\} and D(\BbbZ 4,+4,\Gamma ) = \{ d1, d2, d3, d4\} .
Example 2.4. Let R = \BbbZ 6, \Gamma =
\bigl\{
0, 3
\bigr\}
and I =
\bigl\{
0, 2, 4
\bigr\}
. Then (R,+6,\Gamma ) is a multiplicative
\Gamma -semihyperring with respect to the hyperoperation x\alpha y = x \times 6 \alpha \times 6 y +6 I, for all x, y \in \BbbZ 6
and \alpha \in \Gamma [8]. It is clear that for every \alpha \in \Gamma , we have x\alpha y = I or
\bigl\{
1, 3, 5
\bigr\}
. The function d :
R - \rightarrow R defined by d(x) = 2\times 6 x, for all x \in R, is a derivation.
Example 2.5. Let R = \{ a, b, c\} and \Gamma = \{ \alpha , \beta \} . Then, R is a \Gamma -hyperring with the following
operations and hyperoperation [9]:
\oplus a b c
a a b c
b b \{ a, b\} c
c c c \{ a, b, c\}
\alpha a b c
a a a a
b a b c
c a c b
\beta a b c
a a a a
b a c b
c a b c
It is clear that a is the scaler identity of (R,\oplus ). By the above tables and the conditions of Defini-
tion 2.1, we conclude that there exists only one derivation on R and it is as d1(x) = a, for all x \in R.
By direct calculation, there exist only two weak derivations on R, they are as d1 and d2(x) = x, for
all x \in R. Hence, \Delta (R,+,\Gamma ) = \{ d1\} and D(R,+,\Gamma ) = \{ d1, d2\} .
Example 2.6. Let (R,+) be a group and (\Gamma ,+) be a subgroup of R. Then R is a multiplicative
\Gamma -semihyperring with respect the following hyperoperation x\alpha y = R [9]. It is clear that every
homomorphism on (R,+) is a weak derivation. Therefore, in this case D(R,+,\Gamma ) = \mathrm{H}\mathrm{o}\mathrm{m}(R,+)
which implies that every isomorphism on (R,+) is a derivation.
Example 2.7. Let (R,+, \cdot ) be a commutative ring, (\Gamma ,+) be a subgroup of R and \rho be an
equivalence relation defined as x\rho y if and only if x = y \mathrm{o}\mathrm{r} x = - y. Then the set R/\rho =
\bigl\{
\rho (x) |
x \in R
\bigr\}
becomes a \Gamma -hyperring with respect to the hyperoparation \rho (x)\oplus \rho (y) =
\bigl\{
\rho (x+y), \rho (x - y)
\bigr\}
and multiplication \rho (x)\rho (\alpha )\rho (y) = \rho (x\alpha y), where \Gamma =
\bigl\{
\rho (\alpha ) | \alpha \in \Gamma
\bigr\}
[9]. Suppose that d is a
derivation on (R,+,\Gamma ). We define d on (R/\rho ,\oplus ,\Gamma ) as d(\rho (x)) = \rho (d(x)), for all x \in R. It is clear
that d is well-defined. Also, d is a weak derivation because:
d(\rho (x)\oplus \rho (y)) = d
\bigl\{
\rho (x+ y), \rho (x - y)
\bigr\}
=
\bigl\{
\rho
\bigl(
d(x+ y)
\bigr)
, \rho
\bigl(
d(x - y)
\bigr) \bigr\}
=
=
\Bigl\{
\rho
\bigl(
d(x) + d(y)
\bigr)
, \rho
\bigl(
d(x) - d(y)
\bigr) \Bigr\}
= \rho (d(x))\oplus \rho (d(y)) =
= d(\rho (x))\oplus d(\rho (y))
and
d
\bigl(
\rho (x)\rho (\alpha )\rho (y)
\bigr)
= d
\bigl(
\rho (x\alpha y)
\bigr)
= \rho
\bigl(
d(x\alpha y)
\bigr)
=
= \rho
\bigl(
d(x)\alpha y + x\alpha d(y)
\bigr)
\subseteq
\Bigl\{
\rho (d(x)\alpha y + x\alpha d(y)), \rho
\bigl(
d(x)\alpha y - x\alpha d(y)
\bigr) \Bigr\}
=
= \rho
\bigl(
d(x)\alpha y
\bigr)
\oplus \rho
\bigl(
x\alpha d(y)
\bigr)
= \rho (d(x))\rho (\alpha )\rho (y)\oplus \rho (x)\rho (\alpha )\rho (d(y)) =
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
DERIVATIONS OF GAMMA (SEMI)HYPERRINGS 1015
= d(\rho (x))\rho (\alpha )\rho (y)\oplus \rho (x)\rho (\alpha )d(\rho (y)).
Note that if d is a weak derivation, then d is a weak derivation, too.
Example 2.8. Let \{ An | n \in \BbbR \} be a family of disjoint set such that
An =
\left\{
\{ 0\} if n = 0,
(0,1) if 0 < n < 1,
[m,m+ 1) if m \leq n < m+ 1.
Then for every x \in \BbbR there exists n \in \BbbR such that x \in An. So, \BbbR is a \BbbR -semihyperring with the
following hyperoperation: x \oplus y = An+m and x\alpha y = An\alpha m, for all x \in An, y \in Am and \alpha \in \BbbR
[9]. The function d(x) = 0, for all x \in \BbbR , is a derivation.
The following example shows that the identity function is not always derivation.
Example 2.9. Let R = \{ An | n \in \BbbZ \} be a family of disjoint set such that An = [n, n + 1).
Then for every x \in R there exists n \in \BbbN such that x \in An. So, R is a \BbbZ -semihyperring with
the hyperoperations x \oplus y = An+m and x\alpha y = An\alpha m, for all x \in An, y \in Am and \alpha \in \BbbZ
[8]. The function d(x) = x, for all x \in R, is not a derivation because for all \alpha \in \BbbZ , x \in An
and y \in Am, where n,m \in \BbbN , we have d(x\alpha y) = d(An\alpha m) = An\alpha m = [n\alpha m,n\alpha m + 1)
but d(x)\alpha y \oplus x\alpha d(y) = An\alpha m \oplus An\alpha m = A2(n\alpha m) = [2(n\alpha m), 2(n\alpha m) + 1). For example,
suppose that x = 1, y = 3 and \alpha = 2. Then x \in A1 and y \in A3. Consequently, we find that
d(x\alpha y) = A6 = [6, 7) but d(x)\alpha y \oplus x\alpha d(y) = A6 + A6 = A12 = [12, 13). Note that d is not a
weak derivation, too.
Lemma 2.1. Let (R,+,\Gamma ) be a \Gamma -semihyperring (\Gamma -hyperring) and d be a weak derivation
on R. Then, for all n \in \BbbN , x, y \in R and \alpha \in \Gamma , we have:
(1) If R is commutative, then d( x\alpha x\alpha . . . \alpha x\underbrace{} \underbrace{}
the number of x=n
) \subseteq n
\bigl(
xn - 1\alpha d(x)
\bigr)
. The equality holds when d
is a derivation.
(2) dn(x\alpha y) \subseteq
\sum n
i=0
\biggl(
n
i
\biggr) \bigl(
dn - i(x)\alpha di(y)
\bigr)
, where dn shows derivation of order n and d0(x) =
= x, for all x \in R. The equality holds when d is derivation.
Proof. The proof follows easily by induction.
Let (R,+,\Gamma ) be a \Gamma -semihyperring with zero (\Gamma -hyperring). Then R is said to be of charac-
teristic n, if n is the smallest positive integer such that 0 \in nx = x+ . . .+ x\underbrace{} \underbrace{}
n
, for all x \in R. If no
such of n exists, R is said to be of characteristic 0.
Example 2.10. \Gamma -semihyperring R in Examples 2.3 and 2.4 are of characteristic 4 and 6, re-
spectively.
Let R be a commutative \Gamma -semihyperring with zero (\Gamma -hyperring) and d be a derivation of R.
If R is of characteristic n, then by the above lemma, 0 \in d( x\alpha x\alpha . . . \alpha x\underbrace{} \underbrace{}
the number of x=n
), for all x \in R.
Definition 2.2. A \Gamma -semihyperring R with zero (\Gamma -hyperring) is called prime if 0 \in x\alpha r\beta y, for
all r \in R and \alpha , \beta \in \Gamma , implies that either x = 0 or y = 0. R is called semiprime if 0 \in x\alpha r\beta x,
for all r \in R and \alpha , \beta \in R, implies that x = 0. Obviously, every prime \Gamma -semihyperring (prime
\Gamma -hyperring, respectively) is a semiprime \Gamma -semihyperring (semiprime \Gamma -hyperring, respectively)
but the converse is not always true.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1016 L. K. ARDEKANI, B. DAVVAZ
Example 2.11. In Example 2.2, (R,+,\Gamma ) is (semi)prime if and only if (R,+, \circ ) is (semi)prime.
In Example 2.3, (R,+,\Gamma ) is not semiprime because 0 \in 3\alpha r\beta 3, for all r \in R and \alpha , \beta \in \Gamma , but
3 \not = 0. So, (R,+,\Gamma ) is not prime. Similarly, one can show that in Example 2.4, (R,+,\Gamma ) is not
semiprime and prime.
Lemma 2.2. Let R be a prime \Gamma -semihyperring with zero (\Gamma -hyperring), I be a non-zero ideal
of R and x, y \in R. Then we have:
(1) If I\alpha x = 0, for all \alpha \in \Gamma (or x\alpha I = 0, for all \alpha \in \Gamma ), then x = 0.
(2) If 0 = x\alpha I\beta y, for all \alpha , \beta \in \Gamma , then x = 0 or y = 0.
(3) If x \in Z and 0 \in x\alpha y, for all \alpha \in \Gamma , then x = 0 or y = 0.
Proof. (1) Suppose that I\alpha x = 0, for all \alpha \in \Gamma . Then u\beta r\alpha x \subseteq I\alpha x = \{ 0\} , for all r \in R,
u \in I and \beta \in \Gamma . Hence, x = 0, since R is prime and I \not = 0. In the case x\alpha I = 0, the proof is
similar.
(2) Assume that x\alpha I\beta y = 0, for all \alpha , \beta \in \Gamma . Then x\alpha I\gamma r\beta y \subseteq x\alpha I\beta y = \{ 0\} , for all r \in R
and \gamma \in \Gamma . This implies that x\alpha I\gamma r\beta y = 0, for all r \in R and \alpha , \beta , \gamma \in \Gamma . Hence, x\alpha I = 0 or
y = 0, since R is prime. So, by part (1), x = 0 or y = 0.
(3) Suppose that x \in Z and 0 \in x\alpha y, for all \alpha \in \Gamma . Then, for all r \in R and \beta \in \Gamma , we get
0 \in r\beta 0 = r\beta x\alpha y = x\beta r\alpha y. Therefore, x = 0 or y = 0, since R is prime.
If R is a \Gamma -hyperring, the proof is similar.
Theorem 2.1. Let (R,+,\Gamma ) be a prime \Gamma -semihyperring with zero (\Gamma -hyperring), d be a
derivation on R and I be a non-zero ideal of R. Then, for all x \in R and \alpha \in \Gamma , we have:
(1) If d(I) = 0, then d = 0.
(2) If d(I)\alpha x = 0, for all \alpha \in \Gamma (or x\alpha d(I) = 0, for all \alpha \in \Gamma ), then x = 0 or d = 0.
(3) If d(R)\alpha x = 0, for all \alpha \in \Gamma (or x\alpha d(R) = 0, for all \alpha \in \Gamma ), then x = 0 or d = 0.
Proof. Suppose that (R,+,\Gamma ) is a \Gamma -semihyperring with zero.
(1) For all u \in I, x \in R and \alpha \in \Gamma , we have 0 = d(u\alpha x) = d(u)\alpha x+u\alpha d(x) \supseteq 0+u\alpha d(x) \supseteq
\supseteq u\alpha d(x). Therefore, u\alpha d(x) = 0, for all \alpha \in \Gamma . Consequently, I\alpha d(x) = 0 which implies that
d = 0, by Lemma 2.2 (1).
(2) Suppose that d(I)\alpha x = 0, for all \alpha \in \Gamma . Then 0 = d(y\beta u)\alpha x = d(y)\beta u\alpha x+ y\beta d(u)\alpha x \supseteq
\supseteq d(y)\beta u\alpha x, for all u \in I, y \in R and \beta \in \Gamma . Thus, d(y)\beta u\alpha x = 0, for all \alpha , \beta \in \Gamma . Therefore,
d(y)\beta I\alpha x = 0 which implies that d = 0 or x = 0, by Lemma 2.2 (2). In the case x\alpha d(I) = 0, the
proof is similar.
(3) In (2), put R instead of I.
If R is a \Gamma -hyperring, the proof is similar.
Definition 2.3. Let R be a \Gamma -semihyperring with zero (\Gamma -hyperring). Then R is called n-
torsion free if 0 \in nx = x+ . . .+ x\underbrace{} \underbrace{}
n
, x \in R, implies that x = 0, where n is a positive integer
number.
Example 2.12. In Example 2.3, (R,+,\Gamma ) is 3-torsion free but R is not a 2-torsion free \Gamma -
semihyperring, since 2 + 2 = 0 but 2 \not = 0. In Example 2.4, (R,+,\Gamma ) is not 2-torsion, 3-torsion and
4-torsion free but it is 5-torsion.
Theorem 2.2. Let (R,+,\Gamma ) be a prime 2-torsion free \Gamma -semihyperring with zero (\Gamma -hyperring)
and I be a non-zero ideal of R. Then we have:
(1) If d is a derivation of R such that d(2)(I) = 0, then d = 0.
(2) If d1 and d2 are derivations of R such that d1d2(I) = 0 and d2(I) \subseteq I, then d1 = 0 or
d2 = 0.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
DERIVATIONS OF GAMMA (SEMI)HYPERRINGS 1017
Proof. (1) By Lemma 2.1 for all u, v \in I and \alpha \in \Gamma , we have
0 = d(2)(u\alpha v) = d(2)(u)\alpha v + 2d(u)\alpha d(v) + u\alpha d(2)(v) \supseteq 2d(u)\alpha d(v).
Hence, d(u)\alpha d(v) = 0, since R is 2-torsion free \Gamma -semihyperring. Therefore, d = 0, by Theo-
rem 2.1 (1) and (2).
(2) For all u, v \in I and \alpha \in \Gamma , we get
0 = d1d2(u\alpha v) = d1
\bigl(
d2(u)\alpha v + u\alpha d2(v)
\bigr)
=
= d1d2(u)\alpha v + d2(u)\alpha d1(v) + d1(u)\alpha d2(v) + u\alpha d1d2(v) \supseteq
\supseteq d2(u)\alpha d1(v) + d1(u)\alpha d2(v).
Consequently, d2(u)\alpha d1(v) + d1(u)\alpha d2(v) = 0. By replacing u by d2(u) in the above equation, we
find that d(2)2 (u)\alpha d1(v) \subseteq d
(2)
2 (u)\alpha d1(v) + d1d2(u)\alpha d2(v) = 0, that is d
(2)
2 (u)\alpha d1(v) = 0. Thus,
d1 = 0 or d(2)2 (I) = 0, by Theorem 2.1 (1) and (2). Now, Part (1) implies that d1 = 0 or d2 = 0.
If R is a \Gamma -hyperring, the proof is similar.
Theorem 2.3. Let R be a prime 2-torsion free \Gamma -hyperring, I be a non-zero right ideal of R
and d be a non-zero derivation of R. Then the \Gamma -subhyperring of R generated by d(I) contains no
non-zero right ideal of R if and only if d(I)\Gamma I = 0.
Proof. Suppose that A is a \Gamma -subsemihyperring of R generated by d(I) i.e., A =
\bigl\langle
d(I)
\bigr\rangle
and
set S = A \cap I . For every u \in I, s \in S and \gamma \in \Gamma , we have d(s)\gamma u + s\gamma d(u) = d(s\gamma u) \subseteq A.
Consequently, d(s)\gamma u \subseteq A, I which implies that d(s)\gamma u \subseteq S. It is clear that for every s \in S and
\gamma \in \Gamma , d(s)\gamma I is a right ideal of R. So, d(s)\gamma I =
\bigl\langle
0
\bigr\rangle
= 0, for all s \in S and \gamma \in \Gamma , by hypothesis.
In other hands, d(u\beta a) = d(u)\beta a+ u\beta d(a) \subseteq A, for all u \in I, a \in A, \beta \in \Gamma . Thus, u\beta d(a) \subseteq A.
This implies that u\beta d(a) \subseteq S. Therefore, d
\bigl(
u\beta d(a)
\bigr)
\gamma v = 0, for all v \in I. By Lemma 2.2, 0 =
= d
\bigl(
u\beta d(a)
\bigr)
= d(u)\beta d(a) + u\beta d2(a), for all u \in I, a \in A, \beta \in \Gamma . Replace u by u\gamma v, where
v \in I and \gamma \in \Gamma . Then we get d(u)\gamma v\beta d(a) = 0 which means that d(u)\gamma I\beta d(a) = 0. This implies
that d(u)\gamma I\alpha R\beta d(a)\eta u = 0, for all \gamma , \alpha , \beta , \eta \in \Gamma . Hence, d(u)\alpha I = 0 or d(a)\eta u = 0, since R is
prime. Set L =
\bigl\{
u \in I | d(u)\alpha I = 0, for all\alpha \in \Gamma
\bigr\}
and K =
\bigl\{
u \in I | d(a)\eta u = 0, for all a \in
\in A and \eta \in \Gamma
\bigr\}
. It is clear that L and K are canonical subgroups of I and I = L \cup K. But, I can
not be union of proper canonical subgroups of itself. So, I = L or I = K. If I = K, then we have
d(A)\Gamma I = 0. Hence, d2(I)\Gamma I = 0, since A =
\bigl\langle
d(I)
\bigr\rangle
. By Lemma 2.1, for all u, v, w \in I and \beta , \gamma \in
\Gamma , we obtain 0 = d2(u\beta v)\gamma w = 2d(u)\beta d(v)\gamma w. This implies that d(u)\beta d(v)\gamma w = 0, because R
is 2-torsion free. Replacing u by u\alpha x, where x \in I and \alpha \in \Gamma , we get d(u)\alpha x\beta d(v)\gamma w = 0. So,
d(u)\alpha x\eta R\beta d(v)\gamma w = 0, for all \alpha , \eta , \beta , \gamma \in \Gamma . Hence d(u)\alpha v = 0, for all u, v \in I and \alpha \in \Gamma ,
since R is prime. Therefore, in both of cases we have d(I)\Gamma I = 0.
Conversely, suppose that d(I)\Gamma I = 0. Then A\Gamma I = 0. Hence, A contains no non-zero right
ideal, since R is prime.
A proper ideal P of \Gamma -hyperring R is called prime if for every x, y \in R, x\Gamma R\Gamma y \subseteq P implies
that x \in P or y \in P. This condition is equivalent to I\Gamma J \subseteq P implies that I \subseteq P or J \subseteq P, where
I and J are two ideals of R [5]. A \Gamma -hyperring with \Delta (D, respectively) is called a differential
\Gamma -hyperring (weak differential \Gamma -hyperring, respectively). An ideal I of (weak) differential \Gamma -
hyperring R is called (weak) differential ideal if for all (weak) derivation d of R, we have d(u) \in I,
for all u \in I. For every differential \Gamma -hyperring R,
\bigl\langle
0
\bigr\rangle
R
is a differential ideal.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1018 L. K. ARDEKANI, B. DAVVAZ
Theorem 2.4. Let R be a differential \Gamma -hyperring and P is a prime differential ideal of R.
Then J =
\bigl\{
a \in R | R\Gamma
\sum
a \subseteq P
\bigr\}
is the largest differential ideal of R that is contained in P.
Proof. It is easy to check that J is the largest ideal of R that is contained in P. We prove that
J is differentiable. If a \in J, then R\Gamma
\sum
a \subseteq P, i.e., A =
\Bigl\{
t
\bigm| \bigm| \bigm| t \in
\sum n
i=1
ri\alpha ia, ri \in R,\alpha i \in
\in \Gamma , n \in \BbbN
\Bigr\}
\subseteq P. Therefore, for every
\sum n
i=1
ri\alpha ia \subseteq A, we have d
\Bigl( \sum n
i=1
ri\alpha ia
\Bigr)
\subseteq d(P ) \subseteq P.
Thus,
\sum n
i=1
\bigl(
d(ri)\alpha ia+ ri\alpha id(a)
\bigr)
\subseteq P which leads to
\sum n
i=1
d(ri)\alpha ia +
\sum n
i=1
ri\alpha id(a) \subseteq P.
Then
\sum n
i=1
ri\alpha id(a) \subseteq P which implies that R\Gamma
\sum
d(a) =
\Bigl\{
t
\bigm| \bigm| \bigm| t \in \sum n
i=1
ri\alpha id(a), ri \in R,\alpha i \in
\in \Gamma , n \in \BbbN
\Bigr\}
\subseteq P. Consequently, d(a) \in J.
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Received 25.02.14,
after revision — 15.04.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
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| id | umjimathkievua-article-1613 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:07Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-16132019-12-05T09:21:04Z Derivations of gamma (semi)hyperrings Похiднi гамма (напiв)гiперкiлець Ardekani, L. K. Davvaz, B. Ардекані, Л. К. Давваз, Б. Differential $\Gamma$ -(semi)hyperrings are $\Gamma$ -(semi)hyperrings equipped with derivation, which is a linear unary function satisfying the Leibniz product rule.We introduce the notions of derivation and weak derivation on $\Gamma$ -hyperrings and $\Gamma$ -semihyperrings and obtain some important results relating to them in a specific way. Диференцiальнi $\Gamma$ -(напiв)гiперкiльця — це $\Gamma$-(напiв)гiперкiльця з похiдною, що є лiнiйною унарною функцiєю i задовольняє правило добутку Лейбнiца. Введено поняття похiдної та слабкої похiдної на $\Gamma$ -гiперкiльцях та $\Gamma$ - напiвгiперкiльцях та отримано деякi важливi результати, щo стосуються цих понять у конкретному виглядi. Institute of Mathematics, NAS of Ukraine 2018-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1613 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 8 (2018); 1011-1018 Український математичний журнал; Том 70 № 8 (2018); 1011-1018 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1613/595 Copyright (c) 2018 Ardekani L. K.; Davvaz B. |
| spellingShingle | Ardekani, L. K. Davvaz, B. Ардекані, Л. К. Давваз, Б. Derivations of gamma (semi)hyperrings |
| title | Derivations of gamma (semi)hyperrings |
| title_alt | Похiднi гамма (напiв)гiперкiлець |
| title_full | Derivations of gamma (semi)hyperrings |
| title_fullStr | Derivations of gamma (semi)hyperrings |
| title_full_unstemmed | Derivations of gamma (semi)hyperrings |
| title_short | Derivations of gamma (semi)hyperrings |
| title_sort | derivations of gamma (semi)hyperrings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1613 |
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