Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 625968, 14 pages
doi:10.1155/2012/625968
Research Article
On Prime-Gamma-Near-Rings with
Generalized Derivations
Kalyan Kumar Dey,1 Akhil Chandra Paul,1
and Isamiddin S. Rakhimov2
1
2
Department of Mathematics, Rajshahi University, Rajshahi 6205, Bangladesh
Department of Mathematics, Faculty of Science and Institute for Mathematical Research (INSPEM),
Universiti Putra Malaysia, 43400 Serdang, Malaysia
Correspondence should be addressed to Kalyan Kumar Dey, kkdmath@yahoo.com
Received 26 January 2012; Accepted 19 March 2012
Academic Editor: B. N. Mandal
Copyright q 2012 Kalyan Kumar Dey et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let N be a 2-torsion free prime Γ-near-ring with center ZN. Let f, d and g, h be two
generalized derivations on N. We prove the following results: i if fx, yα 0 or fx, yα ±x, yα or f 2 x ∈ ZN for all x, y ∈ N, α ∈ Γ, then N is a commutative Γ-ring. ii If a ∈ N
and fx, aα 0 for all x ∈ N, α ∈ Γ, then da ∈ ZN. iii If fg, dh acts as a generalized
derivation on N, then f 0 or g 0.
1. Introduction
The derivations in Γ-near-rings have been introduced by Bell and Mason 1. They studied
basic properties of derivations in Γ-near-rings. Then Aşci 2 obtained commutativity
conditions for a Γ-near-ring with derivations. Some characterizations of Γ-near-rings and
regularity conditions were obtained by Cho 3. Kazaz and Alkan 4 introduced the notion
of two-sided Γ-α-derivation of a Γ-near-ring and investigated the commutativity of a prime
and semiprime Γ-near-rings. Uçkun et al. 5 worked on prime Γ-near-rings with derivations
and they found conditions for a Γ-near-ring to be commutative. In 6 Dey et al. studied
commutativity of prime Γ-near-ring with generalized derivations.
In this paper, we obtain the conditions of a prime Γ-near-ring to be a commutative Γring. If a ∈ N, and fx, aα 0 for all x ∈ N, α ∈ Γ, then d is central. Also we prove that if
fg, dh is the generalized derivation on N, then f and g are trivial.
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2. Preliminaries
A Γ-near-ring is a triple N, , Γ, where
i N, is a group not necessarily abelian;
ii Γ is a nonempty set of binary operations on N such that for each α ∈ Γ, N, , α is
a left near-ring;
iii xαyβz xαyβz, for all x, y, z ∈ N and α, β ∈ Γ.
We will use the word Γ-near-ring to mean left Γ-near-ring. For a near-ring N, the set
N0 {x ∈ N : 0αx 0, α ∈ Γ} is called the zero-symmetric part of N. A Γ-near-ring N is said
to be zero-symmetric if N N0 . Throughout this paper, N will denote a zero symmetric left
Γ-near-ring with multiplicative centre ZN. Recall that a Γ-near-ring N is prime if xΓNΓy 0 implies x 0 or y 0. An additive mapping d : N → N is said to be a derivation
on N if dxαy xαdy dxαy for all x, y ∈ N, α ∈ Γ, or equivalently, as noted in
1, that dxαy dxαy xαdy for all x, y ∈ N, α ∈ Γ. Further, an element x ∈ N for
which dx 0 is called a constant. For x, y ∈ N, α ∈ Γ, the symbol x, yα will denote the
commutator xαy − yαx, while the symbol x, y will denote the additive-group commutator
x y − x − y. An additive mapping f : N → N is called a generalized derivation if there exits
a derivation d of N such that fxαy fxαy xαdy for all x, y ∈ N, α ∈ Γ. The concept
of generalized derivation covers also the concept of a derivation.
3. Derivations on Γ-Near-Rings
In this section we prove that a few subsidiary results Lemmas 3.1, 3.2, 3.4, 3.8, 3.9, 3.10 and
3.11 to use them for proving of our main results Theorems 3.3, 3.5, 3.6, 3.12 and 3.13.
Lemma 3.1. Let d be an arbitrary derivation on a Γ-near-ring N. Then N satisfies the following
partial distributive law: aαdb daαbβc aαdbβc daαbβc and daαb aαdbβc daαbβc aαdbβc for all a, b, c ∈ N, α, β ∈ Γ.
Proof. For all a, b, c ∈ N, α, β ∈ Γ, we get daαbβc aαbβdc aαdb daαbβc
and daαbβc aαdbβc daαbβc aαbβdc dbβc daαbβc aαbβdc aαdbβc daαbβc. Equating these two relations for daαbβc now yields the required
partial distributive law.
Lemma 3.2. Let d be a derivation on a Γ-near-ring N and suppose u ∈ N is not a left zero divisor. If
u, duα 0, α ∈ Γ, then x, u is a constant for every x ∈ N.
Proof. From uαu x uαu uαx, for all x ∈ N, α ∈ Γ, we obtain uαdu x duαu x uαdu duαu uαdx duαx, which reduces uαdx duαu duαu uαdx, for
all α ∈ Γ.
Since duαu uαdu, α ∈ Γ, this equation is expressible as uαdx du − dx −
du 0 uαdx, u. Thus dx, u 0.
Theorem 3.3. Let N be a Γ-near-ring having no nonzero divisors of zero. If N admits a nontrivial
commuting derivation d, then N, is abelian.
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Proof. Let c be any additive commutator. Then c is a constant by Lemma 3.2. Moreover, for
any w ∈ N, α ∈ Γ, wαc is an additive commutator, hence also a constant. Thus, 0 dwαc wαdc dwαc and dwαc 0, for all α ∈ Γ. Since dw /
0 for all w ∈ N, we conclude
that c 0.
Lemma 3.4. Let N be a prime Γ-near-ring.
i If z ∈ ZN − {0}, then z is not a zero divisor in N.
ii If ZN − {0} contains an element z for which z z ∈ ZN, then N, is abelian.
iii Let d be a nonzero derivation on N. Then xΓdN {0} implies x 0, and dNΓx {0}
implies x 0.
iv If N is 2-torsion free and d is a derivation on N such that d2 0, then d 0.
Proof. i If z ∈ ZN − {0} and zαx 0, x ∈ N, α ∈ Γ, then zαrβx 0, x, r ∈ N, α ∈ Γ. Thus
we get zΓNΓx 0, by primeness of N, x 0.
ii Let z ∈ ZN − {0} be an element such that z z ∈ ZN, and let x, y ∈ N, α ∈ Γ.
Since z z is distributive, we get x yαz z xαz z yαz z xαz xαz yαz yαz zαx x y y.
On the other hand, x yαz z x yαz x yαz zαx y x y. Thus,
x x y y x y x y and therefore x y y x. Hence N, is abelian.
iii Let xΓdN 0, and let r, s be arbitrary elements of N and α, β ∈ Γ. Then
0 xαdrβs xαrβds xαdrβs xαrβds. Thus xΓNΓdN {0}, and since
dN /
{0}, x 0.
A similar argument works if dNΓx {0}, since Lemma 3.1 provides enough
distributivity to carry it through.
iv For arbitrary x, y ∈ N, α ∈ Γ, we have 0 d2 xαy dxαdy dxαy 2
xαd y dxαdy dxαdy d2 xαy 2dxαdy. Since N is 2-torsion free,
dxαdy 0, x, y ∈ N, α ∈ Γ. Thus dxΓdN {0} for each x ∈ N, and iii yields
da 0. Thus d 0.
Theorem 3.5. If a prime Γ-near-ring N admits a nontrivial derivation d for which dN ∈ ZN,
then N, is abelian. Moreover, if N is 2-torsion free, then N is a commutative Γ-ring.
Proof. Let c be an arbitrary constant, and let x be anon-constant. Then dxαc xαdc dxαc dxαc ∈ ZN, α ∈ Γ. Since dx ∈ ZN − {0}, it follows easily that c ∈ ZN.
Since c c is a constant for all constants c, it follows from Lemma 3.4ii that N, is abelian,
provided that there exists a nonzero constant.
Assume, then, that 0 is the only constant. Since d is obviously commuting, it follows
from Lemma 3.2 that all u which are not zero divisors belong to the center of N, , denoted
by ZN. In particular, if x /
0, dx ∈ ZN. But then for all y ∈ N, dydx−dy−dx dy, x 0, hence y, x 0.
Now we assume that N is 2-torsion free. By Lemma 3.1, aαdb daαbβc aαdbβc daαbβc for all a, b, c ∈ N, α, β ∈ Γ, and using the fact that daαb ∈ ZN,
α ∈ Γ, we get cαaβdb cαdaβb aαdbβc aαdbβc, α, β ∈ Γ. Since N, is abelian
and dN ⊆ ZN, this equation can be rearranged to yield dbαc, aβ daαb, cβ for all
a, b, c ∈ N, α, β ∈ Γ.
Suppose now that N is not commutative. Choosing b, c ∈ N, with b, cβ / 0, β ∈ Γ,
2
and letting a dx, we get d xαb, cβ 0, for all x ∈ N, α, β ∈ Γ, and since the central
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elements d2 x cannot be a nonzero divisor of zero, we conclude that d2 x 0 for all x ∈ N.
But by Lemma 3.4iv, this cannot happen for nontrivial d.
Theorem 3.6. Let N be a prime Γ-near-ring admitting a nontrivial derivation d such that
dx, dyα 0 for all x, y ∈ N, α ∈ Γ. Then N, is abelian. Moreover, if N is 2-torsion
free, then N is a commutative Γ-ring.
Proof. By Lemma 3.4ii, if both z and z z commute element-wise with dN, then zαdc 0, α ∈ Γ, for all additive commutators c. Thus, taking z dx, we get dxαdc 0 for all
x ∈ N, α ∈ Γ, so dc 0 by Lemma 3.4iii. Since wαc is also an additive commutator for
any w ∈ N, α ∈ Γ, we have dwαc 0 dwαc, and another application of Lemma 3.4iii
gives c 0.
Now we assume that N is 2-torsion free. By the partial distributive law,
ddxαyβdz dxαdyβdz d2 xαyβdz for all x, y, z ∈ N, α, β ∈ Γ, hence,
d2 xαyβdz ddxαyβdz − dxαdyβdz dxαdyβdz dzαddxβy −
dxβdy dzαd2 xβy d2 xαdzβy, α, β ∈ Γ. Thus d2 xαyβdz − dzβy 0 for
all x, y z ∈ N, α, β ∈ Γ.
Replacing yδt, δ ∈ Γ, we obtain d2 xαyδtβdz d2 xαdzβyδt d2 xαyβdzδt,
for all x, y, z, t ∈ N, α, β, δ ∈ Γ, so that d2 xαyβt, dzδ 0 for all x, y, z, t ∈ N, α, β, δ ∈ Γ.
The primeness of N shows that either d2 0 or dN ⊆ ZN, and since the first of these
conditions is impossible by Lemma 3.4iv, the second must hold and N is a commutative
Γ-ring by Theorem 3.5.
Definition 3.7. Let N be a Γ-near-ring and d a derivation of N. An additive mapping f : N →
N is said to be a right generalized derivation of N associated with d if
f xαy fxαy xαd y
∀x, y ∈ R, α ∈ Γ,
3.1
and f is said to be a left generalized derivation of N associated with d if
f xαy dxαy xαf y
∀x, y ∈ R, α ∈ Γ.
3.2
Finally, f is said to be a generalized derivation of N associated with d if it is both a left and
right generalized derivation of N associated with d.
Lemma 3.8. Let f be a right generalized derivation of a Γ-near ring N associated with d. Then
i fxαy xαdy fxαy for all x, y ∈ N, α ∈ Γ;
ii fxαy xαfy dxαy for all x, y ∈ N, α ∈ Γ.
Proof. i For any x, y ∈ N, α ∈ Γ, we get
f xα y y fxα y y xαd y y fxαy fxαy xαd y xαd y ,
f xαy xαy fxαy xαd y fxαy xαd y .
3.3
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Comparing these two expressions, we obtain
fxαy xαd y xαd y fxαy
∀x, y ∈ N, α ∈ Γ,
3.4
and so,
f xαy xαd y fxαy
∀x, y ∈ N, α ∈ Γ.
3.5
ii In a similar way.
Lemma 3.9. Let f be a right generalized derivation of a Γ-near ring N associated with d. Then
i fxαy xαdyβz fxαyβz xαdyβz, for all x, y ∈ N, α, β ∈ Γ.
ii dxαy xαfyβz dxαyβz xαfyβz, for all x, y ∈ N, α, β ∈ Γ.
Proof. i For any x, y, z ∈ N, α, β ∈ Γ, we get fxαyβz fxαyβz xαyβdz.
On the other hand,
f xα yβz fxαyβz xαd yβz fxαyβz xαd y βz xαyβdz.
3.6
From these two expressions of fxαyβz, we obtain that, for all x, y, z ∈ N, α, β ∈ Γ,
fxαy xαd y βz fxαyβz xαd y βz.
3.7
ii The proof is similar.
Lemma 3.10. Let N be a prime Γ-near-ring, f a nonzero generalized derivation of N associated with
the nonzero derivation d and a ∈ N. i If aΓfN 0, then a 0. ii If fNΓa 0, then a 0.
Proof. i For any x, y ∈ N, α, β ∈ Γ, we get 0 aβfxαy aβfxαyaβxαdy aβxαdy.
Hence aΓNΓdN 0. Since N is a prime Γ-near-ring and d /
0, we obtain a 0.
ii A similar argument works if fNΓa 0.
Lemma 3.11. Let N be a prime Γ-near-ring. Let f be a generalized derivation of N associated with
the nonzero derivation d. If N is a 2-torsion free Γ-near-ring and f 2 0, then f 0.
Proof. i For any x, y ∈ N, α ∈ Γ, we get
0 f 2 xαy f f xαy f fxαy xαd y f 2 xαy 2fxαd y xαd2 y .
3.8
By the hypothesis,
2fxαd y xαd2 y 0
∀x, y ∈ N, α ∈ Γ.
Writing fx by x in 3.9, we get fxαd2 y 0, for all x, y ∈ N, α ∈ Γ.
3.9
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By Lemma 3.9ii, we obtain that d2 N 0 or f 0. If d2 N 0 then d 0 from
Lemma 3.4iv, a contradiction. So we find f 0.
Theorem 3.12. Let N be a prime Γ-near-ring with a nonzero generalized derivation f associated with
d. If fN ⊆ ZN, then N, is abelian. Moreover, if N is 2-torsion free, then N is commutative
Γ-ring.
Proof. Suppose that a ∈ N, such that fa /
0. So, fa ∈ ZN − {0} and fa fa ∈
ZN − {0}. For all x, y ∈ N, α ∈ Γ, we have x yαfa fa fa faαx y.
That is, xαfa xαfa yαfa yαfa faαx faαx faαy faαy, for
all x, y ∈ N, α ∈ Γ.
Since fa ∈ ZN, we get faαx faαy faαy faαx, and so, faαx, y 0
for all x, y ∈ N, α ∈ Γ.
Since fa ∈ ZN − {0} and N is a prime Γ-near-ring, it follows that x, y 0, for all
x, y ∈ N. Thus N, is abelian.
Using the hypothesis, for any x, y, z ∈ N, α, β ∈ Γ, zαfxβy fxβyαz. By
Lemma 3.4ii, we have zαdxβy zαxfy dxαyβz xαfyβz. Using fN ⊂ ZN
and N, being abelian, we obtain that
zαdxβy − dxαyβz x, zα βf y ,
∀x, y ∈ N, α, β ∈ Γ.
3.10
Substituting fz for z in 3.10, we get fzβdx, yα 0 for all x, y ∈ N, α, β ∈ Γ.
Since fz ∈ ZN and f a nonzero generalized derivation with associated with d, we
get dN ⊂ ZN. So, N is a commutative Γ-ring by Theorem 3.3.
Theorem 3.13. Let N be a prime Γ-near-ring with a nonzero generalized derivation f associated with
d. If fN, fNα 0, α ∈ Γ, then N, is abelian. Moreover, if N is 2-torsion free, then N is
commutative Γ-ring.
Proof. By the same argument as in Theorem 3.12, it is shown that if both z and z z commute
elementwise with fN, then we have
zαf x, y 0 ∀x, y ∈ N, α ∈ Γ.
3.11
Substituting ft, t ∈ N for z in 3.11, we get ftαfx, y 0, α ∈ Γ. By Lemma 3.9i,
we obtain that fx, y 0 for all x, y ∈ N, α ∈ Γ. For any w ∈ N, β ∈ Γ, we have 0 fwβx, wβy fwβx, y dwβx, y wβfx, y and so, we obtain dwβx, y 0, for
any w ∈ N, β ∈ Γ. From Lemma 3.4iii, we get x, y 0 for any x, y ∈ N.
Now we assume that N is 2-torsion free. By the assumption fN, fNα 0, α ∈ Γ,
we have
fzαf fxβy f fxβy αfz
∀x, y, z ∈ N, α, β ∈ Γ.
3.12
Hence we get
fzαd fx βy fzαfxβf y d fx αyβfz fxαf y βfz,
fzαd fx βy fxαfzβf y d fx αyβfz fxαfzβf y ,
3.13
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and so,
fzαd fxβy d fx αyβfz
∀x, y, z ∈ N, α, β ∈ Γ.
3.14
If we take yδw instead of y in 3.14, then
d fx αyδwβfz fzαd fxβyδw d fx αyδfzβw
∀x, y, z ∈ N, α, β, δ ∈ Γ,
3.15
and so,
d fx αyδwβfz − d fx αyδfzβw d fx αyδ fz, w δ 0
∀x, y, z ∈ N, α, β, δ ∈ Γ.
3.16
Thus we get dfxΓNΓfz, wδ 0, for all x, y, z ∈ N, α, β, δ ∈ Γ. Since N is a prime
Γ-near-ring, we have dfN 0 or fN ⊂ ZN. Let us assume that dfN 0. Then
0 d f xαy d dxαy xαf y
3.17
and so,
d2 xαy dxαd y dxαf y 0,
∀x, y ∈ N, α ∈ Γ.
3.18
Replacing y by yβz, β ∈ Γ, in 3.18, we get
0 d2 xαyβz dxαd yβz dxαf yβz
d2 xαyβz dxαd y βz dxαyβdz dxαf y βz dxαyβdz
d2 xαy dxαd y dxαf y βz 2dxαyβdz ∀x, y, z ∈ N, α, β ∈ Γ.
3.19
Using 3.18 and N being 2-torsion free Γ-near-ring, we get dNΓNΓdN 0.
Thus we obtain that d 0. It contradicts by d /
0. The theorem is proved.
4. Generalized Derivations of Γ-Near-Rings
We denote a generalized derivation f : N → N determined by a derivation d of N by f, d.
We assume that d is a nonzero derivation of N unless stated otherwise.
Theorem 4.1. Let f, d be a generalized derivation of N. If fx, yα 0 for all x, y ∈ N, α ∈ Γ,
then N is a commutative Γ-ring.
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Proof. Assume that fx, yα 0 for all x, y ∈ N, α ∈ Γ. Substitute xβy instead of y, obtaining
f
x, xβy α f xβ x, y α dxβ x, y α xβf x, y α 0.
4.1
Since the second term is zero, it is clear that
dxαxβy dxαyβx
∀x, y ∈ N, α, β ∈ Γ.
4.2
Replacing y by yδz in 4.2 and using this equation, we get
dxαyβx, zδ 0 ∀x, y, z ∈ N, α, β, δ ∈ Γ.
4.3
Hence either x ∈ ZN or dx 0. Let L {x ∈ N | dx 0}. Then ZN and L are two
additive subgroups of N, ZN ∪ L. However, a group cannot be the union of proper
subgroups, hence either N ZN or N L. Since d / 0, we are forced to conclude that N is
a commutative Γ-ring.
Theorem 4.2. Let f, d be a generalized derivation of N. If fx, yα ±x, yα for all x, y ∈ N,
α ∈ Γ, then N is a commutative Γ-ring.
Proof. Assume that fx, yα ±x, yα for all x, y ∈ N, α ∈ Γ. Replacing y by xβy, β ∈ Γ, in
the hypothesis, we have
f
x, xβy α ± xαxβy − xαyβx ±xβ x, y α .
4.4
On the other hand,
f
x, xβy α f xβ x, y α dxβ x, y α xβf x, y α dxβ x, y α xβ ± x, y α .
4.5
It follows from the two expressions for fx, xβyα that
dxαxβy dxαyβx
∀x, y ∈ N, α, β ∈ Γ.
4.6
Using the same argument as in the proof of Theorem 4.1, we get that N is a commutative
Γ-ring.
Theorem 4.3. Let f, d be a nonzero generalized derivation of N. If f acts as a homomorphism on
N, then f is the identity map.
Proof. Assume that f acts as a homomorphism on N. Then one obtains
f xαy fxαf y dxαy xαf y
∀x, y ∈ N, α ∈ Γ.
4.7
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Replacing y by yβz in 4.7, we arrive at
fxαf yβz dxαyβz xαf yβz .
4.8
Since f, d is a generalized derivation and f acts as a homomorphism on N, we deduce that
f xαy βfz dxαyβz xαd y βz xαyβfz.
4.9
By Lemma 3.9ii, we get
dxαyβfz xαf y βfz dxαyβz xαd y βz xαyβfz,
4.10
dxαyβfz xαf yβz dxαyβz xαd y βz xαyβfz.
4.11
and so
That is,
dxαyβfz xαd y βz xαyβfz dxαyβz xαd y βz xαyβfz.
4.12
Hence, we deduce that
dxαyβ fz − z 0 ∀x, y, z ∈ N, α, β ∈ Γ.
4.13
Because N is prime and d /
0, we have fz z for all z ∈ N. Thus, f is the identity map.
Theorem 4.4. Let f, d be a nonzero generalized derivation of N. If f acts as an antihomomorphism
on N, then f is the identity map.
Proof. By the hypothesis, we have
f xαy f y αfx dxαy xαf y
∀x, y ∈ N, α ∈ Γ.
4.14
Replacing y by xβy in the last equation, we obtain
f xβy αfx dxβxαy xβf xαy .
4.15
Since f, d is a generalized derivation and f acts as an antihomomorphism on N, we get
dxβy xβf y αfx dxαxβy xαf y βfx.
4.16
By Lemma 3.9ii, we conclude that
dxαyβfx xαf y βfx dxαxβy xαf y βfx,
4.17
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and so
dxαyβfx dxαxβy
∀x, y ∈ N, α, β ∈ Γ.
4.18
Replacing y by yδz and using this equation, we have
dxαyβ fx, z α 0 ∀x, z ∈ N, α, β ∈ Γ.
4.19
Hence we obtain the following alternatives: dx 0 or fx ∈ ZN, for all x ∈ N. By a
standard argument, one of these must hold for all x ∈ N. Since d /
0, the second possibility
gives that N is commutative Γ-ring by Theorem 3.12, and so we deduce that f is the identity
map by Theorem 4.3.
Theorem 4.5. Let f, d be a generalized derivation of N such that dZN /
0, and a ∈ N. If
fx, aα 0 for all x ∈ N, α, β ∈ Γ, then a ∈ ZN.
Proof. Since dZN / 0, there exists c ∈ ZN such that dc / 0. Furthermore, as d is a
derivation, it is clear that dc ∈ ZN. Replacing x by cβx, β ∈ Γ, in the hypothesis and
using Lemma 3.9ii, we have
f cβx αa aαf cβx ,
dcαxβa cαfxβa aαdcβx aαcβfx.
4.20
Since c ∈ ZN and dc ∈ ZN, we get
dcαxβ y, a α 0
∀y ∈ N, α, β, δ ∈ Γ.
4.21
By the primeness of N and 0 / dc ∈ ZN, we obtain that a ∈ ZN.
Theorem 4.6. Let f, d be a generalized derivation of N, and a ∈ N. If fx, aα 0 for all
x ∈ N, then da ∈ ZN.
Proof. If a 0, then there is nothing to prove. Hence, we assume that a /
0.
Replacing x by aβx in the hypothesis, we have
f aβx αa aαf aβx ,
daαxβa aαfxβa aαdaβx aαaβfx.
4.22
Using fxαa aαfx, we have
daαxβa aαdaβx
∀x ∈ N, α, β ∈ Γ.
4.23
Taking xδy instead of x in the last equation and using this, we conclude that
daαNβ a, y α 0
∀y ∈ N, α, β ∈ Γ.
4.24
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Since N is a prime Γ-near-ring, we have either da 0 or a ∈ ZN. If 0 /
a ∈ ZN, then
N, is abelian by Lemma 3.2ii. Thus
fxαa faαx
fxαa xαda daαx aαfx
4.25
and so
da, xα 0 ∀x ∈ N, α ∈ Γ.
4.26
That is, da ∈ ZN. Hence in either case we have da ∈ ZN. This completes the
proof.
Theorem 4.7. Let f, d be a generalized derivation of N. If N is a 2-torsion free Γ-near-ring and
f 2 N ⊂ ZN, then N is a commutative Γ-ring.
Proof. Suppose that f 2 N ⊂ ZN. Then we get
f 2 xαy f 2 xαy 2fxαd y xαd2 y ∈ ZN
∀x, y ∈ N, α ∈ Γ.
4.27
In particular, f 2 xαc 2fxαdc xαd2 c ∈ ZN for all x ∈ N, c ∈ ZN, α ∈ Γ. Since
the first summand is an element of ZN, we have
2fxαdc xαd2 c ∈ ZN ∀x ∈ N, c ∈ ZN, α ∈ Γ.
4.28
Taking fx instead of x in 4.28, we obtain that
2f 2 xαdc fxαd2 c ∈ ZN ∀x ∈ N, c ∈ ZN, α ∈ Γ.
4.29
Since dc ∈ ZN, f 2 x ∈ ZN, and so f 2 xαdc ∈ ZN for all x ∈ N, c ∈ ZN, α ∈ Γ,
we conclude fxαd2 c ∈ ZN for all x ∈ N, c ∈ ZN, α ∈ Γ.
Since N is prime, we get d2 ZN 0 or fN ⊆ ZN. If fN ⊆ ZN, then
N is a commutative Γ-ring by Lemma 3.8. Hence, we assume d2 Z 0. By 4.28, we get
2fxαdc ∈ ZN for all x ∈ N, c ∈ ZN, α ∈ Γ.
Since N is a 2-torsion free near-ring and dc ∈ ZN, we obtain that either fN ⊂
ZN or dZN 0. If fN ⊂ ZN, then we are already done. So, we may assume that
dZN 0. Then
fcαx fxαc,
fcαx cαdx fxαc xαdc,
4.30
and so
fcαx cαdx fxαc
∀x ∈ N, c ∈ ZN.
4.31
12
International Journal of Mathematics and Mathematical Sciences
Now replacing x by fx in 4.31, and using the fact that f 2 N ⊂ ZN, we get
fcαfx cαd fx f 2 xαc
∀x ∈ N, c ∈ ZN.
4.32
∀x ∈ N, c ∈ ZN, α ∈ Γ.
4.33
That is,
fcαfx cαd fx ∈ Z
Again taking fx instead of x in this equation, one can obtain
fcαf 2 x cαd f 2 x ∈ Z
∀x ∈ N, c ∈ ZN, α ∈ Γ.
4.34
The second term is equal to zero because of dZ 0. Hence we have fcαf 2 x ∈ ZN for
all x ∈ N, c ∈ ZN, α ∈ Γ.
Since f 2 N ⊂ ZN by the hypothesis, we get either f 2 N 0 or fZN ⊂ ZN.
2
If f N 0, then the theorem holds by Definition 3.7. If fZ ⊂ ZN, then fxαfc ffcαx for all x ∈ N, c ∈ ZN, and so
dxαfc fcαfx
∀x ∈ N, c ∈ ZN.
4.35
Using fc ∈ ZN, we now have fcαdx − fx 0 for all x ∈ N, c ∈ ZN, α ∈ Γ. Since
fZN ⊂ ZN, we have either fZN 0 or d f. If d f, then f is a derivation of N
and so N is commutative Γ-ring by Lemma 3.11.
Now assume that fZN 0. Returning to the equation 4.31, we have
cα dx − fx 0
∀x ∈ N, c ∈ ZN, α ∈ Γ.
4.36
Since c ∈ ZN, we have either d f or ZN 0. Clearly, d f implies the theorem
holds. If ZN 0, then f 2 N 0 by the hypothesis, and so N is a commutative Γ-ring by
Lemma 3.4iv. Hence, the proof is completed.
Corollary 4.8. Let N be a 2-torsion free near-ring, and f, d a nonzero generalized derivation of N.
If fN, fNα 0, α ∈ Γ, then N is a commutative Γ-ring.
Lemma 4.9. Let f, d and g, h be two generalized derivations of N. If h is a nonzero derivation on
N and fxαhy −gxαdy for all x, y ∈ N, then N, is abelian.
Proof. Suppose that fxαhy gxαdy 0 for all x, y ∈ N, α ∈ Γ.
We substitute y z for y, thereby obtaining
fxαh y fxαhz gxαd y gxαdz 0.
4.37
Using the hypothesis, we get
fxαh y, z 0 ∀x, y, z ∈ N, α ∈ Γ.
4.38
International Journal of Mathematics and Mathematical Sciences
13
It follows by Lemma 3.10ii that hy, z 0 for all y, z ∈ N. For any w ∈ N, we have
hwαy, wαz hwαy, z hwαy, z wαhy, z 0 and so hwαy, z 0 for all
w, y, z ∈ N, α ∈ Γ.
An appeal to Lemma 3.4iii yields that N, is abelian.
Theorem 4.10. Let f, d and g, h be two generalized derivations of N. If N is 2-torsion free and
fxαhy −gxαdy for all x, y ∈ N, α ∈ Γ, then f 0 or g 0.
Proof. If h 0 or d 0, then the proof of the theorem is obvious. So, we may assume that h /
0
and d /
0. Therefore, we know that N, is abelian by Lemma 4.9.
Now suppose that
fxαh y gxαd y 0 ∀x, y ∈ N, α ∈ Γ.
4.39
Replacing x by uβv in this equation and using the hypothesis, we get
f uβv αh y g uβv αd y
uαfvβh y duαvβh y uαgvβd y huαvβd y
4.40
0,
and so
duαvβh y −huαvβd y
∀u, v, y ∈ N, α ∈ Γ.
4.41
Taking yδt instead of y in the above relation, we obtain
duαvβh y δt duαvyβδht −huαvβd y δt − huαvβyδdt.
4.42
That is,
duαvβyδht −huαvβyδdt
∀u, v, y, t ∈ N, α, β, δ ∈ Γ.
4.43
Replacing y by hy in 4.43 and using this relation, we have
huαNβ d y δht − h y αdt 0 ∀u, y, t ∈ N.
4.44
Since N is a prime Γ-near-ring and h /
0, we obtain that
d y αht h y αdt,
∀y, t ∈ N.
4.45
Now again taking uλv instead of x in the initial hypothesis, we get
fuαvβh y uαdvβh y guαvβd y uαhvβd y 0.
4.46
14
International Journal of Mathematics and Mathematical Sciences
Using 4.45 yields that
fuαvβh y 2uαhvβd y guαvβd y 0
∀u, v, y ∈ N,
4.47
Taking hv instead of v in this equation, we arrive at
fuαhvβh y 2uαh2 vβd y guαhvβd y 0.
4.48
By the hypothesis and 4.45, we have
0 −guαdvβh y 2uαh2 vβd y guαhvβd y
−guαhvβd y 2uαh2 vβd y guαhvβd y ,
4.49
and so
2uαh2 vβd y 0
∀u, v, y ∈ N, α, β ∈ Γ.
4.50
Since N is a 2-torsion free prime Γ-near-ring, we obtain that h2 NΓdN 0. An appeal to
Lemma 3.4iii and iv gives that h 0. This contradicts by our assumption. Thus the proof
is completed.
Theorem 4.11. Let f, d and g, h be two generalized derivations of N. If fg, dh acts as a
generalized derivation on N, then f 0 or g 0.
Proof. By calculating fgxαy in two different ways, we see that gxαdy fxαhy 0
for all x, y ∈ N, α ∈ Γ. The proof is completed by using Theorem 4.10.
Acknowledgments
The paper was supported by Grant 01-12-10-978FR MOHE Malaysia. The authors are
thankful to the referee for valuable comments.
References
1 H. E. Bell and G. Mason, “On derivations in near-rings,” in Near-Rings and Near-Fields (Tübingen, 1985),
vol. 137 of North-Holland Mathematics Studies, pp. 31–35, North-Holland, Amsterdam, The Netherlands,
1987.
2 M. Aşci, “Γ-σ, τ derivation on gamma near rings,” International Mathematical Forum. Journal for Theory
and Applications, vol. 2, no. 1–4, pp. 97–102, 2007.
3 Y. U. Cho, “A study on derivations in near-rings,” Pusan Kyongnam Mathematical Journal, vol. 12, no. 1,
pp. 63–69, 1996.
4 M. Kazaz and A. Alkan, “Two-sided Γ-α-derivations in prime and semiprime Γ-near-rings,” Korean
Mathematical Society, vol. 23, no. 4, pp. 469–477, 2008.
5 M. Uçkun, M. A. Öztürk, and Y. B. Jun, “On prime gamma-near-rings with derivations,” Korean Mathematical Society, vol. 19, no. 3, pp. 427–433, 2004.
6 K. K. Dey, A. C. Paul, and I. S. Rakhimov, “Generalized derivations in semiprime gamma rings,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 270132, 14 pages, 2012.
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