On Ideals and Commutativity of Prime Rings with Generalized Derivations

Abstract

An additive mapping F: R → R is called a generalized derivation on R if there exists a derivation d: R → R such that F(xy) = xF(y) + d(x)y holds for all x,y ∈ R. It is called a generalized (α,β)−derivation on R if there exists an (α,β)−derivation d: R → R such that the equation F(xy) = F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisfies certain differential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.

Full text

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 1, 2018, 79-89
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global
On ideals and commutativity of prime rings with
generalized derivations
M. K. Abu Nawas1,∗, Radwan M. Al-Omary2
1Department of Mathematics, Faculty of Science, Northern Border University, Arar,
Saudi Arabia
2Department of Mathematics, Ibb University, Yemen
Abstract. An additive mapping F:R→Ris called a generalized derivation on Rif there exists
a derivation d:R→Rsuch that F(xy) = xF (y) + d(x)yholds for all x, y ∈R. It is called a
generalized (α, β)−derivation on Rif there exists an (α, β)−derivation d:R→Rsuch that the
equation F(xy) = F(x)α(y) + β(x)d(y) holds for all x, y ∈R. In the present paper, we investigate
commutativity of a prime ring R, which satisfies certain differential identities on the left ideals of
R. Moreover some results on commutativity of rings with involutions that satisfy certain identities
are proved.
2010 Mathematics Subject Classifications: 16D90, 16W25, 16N60, 16U80
Key Words and Phrases: Left ideals, prime rings, centralizing, derivations, generalized deriva-
tions, commutativity
1. Introduction
Recently, a considerable number of researchers have investigated the ideals in prime
rings as well as the commutativity of prime rings that consider derivations and generalized
derivations, see for example [2], [3], [5] and [7]. In [4], Ashraf and Khan showed that a
∗-ideal Uis central if the ring Radmits a general derivation Fassociated with a derivation
dsatisfying specific properties. In [10] , El-Soufi and Aboubakr proved that J⊆Z(R)
under specific properties, where Ris a 2-torsion free prime ring with center Z(R) admitting
a generalized derivation Fassociated with a derivation d,Jis a nonzero Jordan ideal. In
addition, Ibraheem in [11] showed that if fis a generalized reverse derivation on Rsuch
that fis commuting and centralizing on a right ideal Iof R, then Ris a commutative,
where Ris a prime ring and dis a reverse derivation on R. Moreover, in [1], Abu Nawas
and Al-Omary investigated the commutativity of Rsuch that Ris a ∗-prime ring admitting
∗Corresponding author.
Email addresses: m.abunawas.math.nbu@gmail.com (M. Abu Nawas), radwan959@yahoo.com (Radwan
Al-Omary)
http://www.ejpam.com 79 c
2018 EJPAM All rights reserved.

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 80
generalized (α,β)- derivations Fand Gassociated with (α, β)−derivations dand g, respec-
tively, that satisfying certain properties. Let Rbe an associative ring with center Z(R).
For x, y ∈Rdenote the commutator xy −yx by [x, y] and the anti-commutator xy +yx
by x◦y. Recall that a ring Ris prime if for any a, b ∈R,aRb ={0}implies that a= 0
or b= 0. An additive mapping d:R−→ Ris called a derivation if d(xy) = d(x)y+xd(y)
for all x, y ∈R. In particular, for a fixed a∈R, the mapping Ia:R−→ Rgiven by
Ia(x) = [x, a] is a derivation called an inner derivation. An additive mapping x7→ x∗on
a ring Ris called an involution if (x∗)∗=xand (xy)∗=y∗x∗for all x, y ∈R. A ring R
equipped with an involution ∗is said to be a ∗-prime ring if aRb =aRb∗={0}implies
a= 0 or b= 0 for any a, b ∈R.
An additive function F:R−→ Ris called a generalized inner derivation if F(x) = ax+xb
for fixed a, b ∈R. For such a mapping F, it is easy to see that
F(xy) = xF (y)+[a, x]y=xF (y) + Ia(x)yfor all x, y ∈R.
This observation leads to the following definition, given in [9]: an additive mapping F:
R−→ Ris called a generalized derivation with associated derivation dif
F(xy) = xF (y) + d(x)yfor all x, y ∈R.
Familiar examples of generalized derivations are derivations and generalized inner deriva-
tions that include left multipliers and right multipliers. Since the sum of two generalized
derivations is a generalized derivation, every map of the form F(x) = xc +d(x), where c
is a fixed element of Rand dis a derivation, is a generalized derivation; and if Rhas 1,
all generalized derivations have this form. Let αand βbe endomorphisms of R. We shall
write for any pair of x, y ∈R, [x, y]α,β =xα(y)−β(y)x, (x◦y)α,β =xα(y) + β(y)xAn
additive map d:R−→ Ris called an (α, β)-derivation if d(xy) = d(x)α(y) + β(x)d(y) for
all x, y ∈R. An additive mapping F:R−→ Ris called a generalized (α, β)-inner deriva-
tion if F(x) = aα(x) + β(x)b, for some fixed a, b ∈Rand for all x∈R. An additive map
F:R−→ Ris called a generalized (α, β)-derivation associated with an (α, β)-derivation
d:R−→ Rif
F(xy) = F(x)α(y) + β(x)d(y) for all x, y ∈R.
Over the last four decade, several authors have proved results on commutativity of
prime rings or semiprime rings that admitting automorphisms, derivations or generalized
derivations which are centralizing or commuting on appropriate subset of R(see [5], [8],
[12] , [14]-[16]).
In this paper, we investigate the commutativity of a prime ring Radmitting generalized
derivations Fand Gsatisfying any one of the following properties: (i)F(x)◦x∈Z(R), (ii)
[F(x), F (y)] −F[x, y]∈Z(R), (iii)F(x)◦F(y)−F(x◦y)∈Z(R), (iv)F[x, y]+[F(x), y]−
[F(x), F (y)] ∈Z(R), (v)F(x◦y)−[x, y]∈Z(R), (vi) [F(x), F (y)] −x◦y∈Z(R), (vii)
[F(x), G(y)] −[x, y]∈Z(R),(viii) [F(x), x]−[x, G(x)] ∈Z(R) and F(x)◦x−x◦G(x)∈
Z(R) for all x, y in some appropriate subset of R. Some results on commutativity of rings
with involutions that satisfy certain identities are also proved.

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 81
2. Preliminaries
We shall use, without explicit mention, the following basic identities that hold for any
x, y, z ∈R:
[xy, z] = x[y, z]+[x, z]y;
[x, yz] = y[x, z]+[x, y]z;
x◦(yz)=(x◦y)z−y[x, z] = y(x◦z)+[x, y]z;
(xy)◦z=x(y◦z)−[x, z]y= (x◦z)y+x[y, z];
[xy, z]α,β =x[y, z]α,β + [x, β(z)]y=x[y, α(z)] + [x, z]α,β y;
[x, yz]α,β =β(y)[x, z]α,β + [x, y]α,β α(z);
(x◦(yz))α,β = (x◦y)α,β α(z)−β(y)[x, z]α,β =β(y)(x◦z)α,β + [x, y]α,β α(z);
((xy)◦z)α,β =x(y◦z)α,β −[x, β(z)]y= (x◦z)α,β y+x[y, α(z)].
The following results are also going to be used:
Remark 2.1.In a prime ring, the centralizer of any nonzero one-sided ideal is equal to the
center of R; in particular, if Rhas nonzero central ideal, Rmust be commutative.
Remark 2.2.Let Rbe a prime ring. For a nonzero element a∈Z(R), if ab ∈Z(R), then
b∈Z(R).
We begin our discussion with the following results.
Lemma 2.1. Let Rbe a prime ring. If d:R−→ Ris a derivation on R, then for any
06=z∈Z(R),d(z)∈Z(R).
Proof. We have 0 6=z∈Z(R), that is [z, r] = 0 for all r∈Rand hence d[z, r] = 0,
d(zr −rz) = 0, i.e d(z)r+z(d(r)−d(r)z−rd(z) = 0, that is [d(z), r]+[z, d(r)] = 0, since
z∈Z(R) so we get [d(z), r] = 0 for all r∈Rwhich yields that d(z)∈Z(R).
Lemma 2.2. [[6], Theorem 2] Let Rbe a prime ring and Ia nonzero left ideal of R
such that I∩Z(R)6= 0. If Radmits a generalized derivation Fwith associated derivation
dsuch that Fis centralizing on I, then Ris commutative.
Lemma 2.3. [[7], Theorem 4] Let Rbe a prime ring and Ia nonzero left ideal. If
Radmits a nonzero derivation dsuch that [d(x), x]∈Z(R)for all x∈I, then Ris
commutative.
Lemma 2.4. [[13], Lemma 2.5] If a prime ring Rcontains a nonzero commutative right
ideal, then Ris commutative.
Lemma 2.5. Let Rbe a prime ring and Ibe a nonzero left ideal of Rsuch that
(a) [x, y]∈Z(R)for all x, y ∈I, or
(b)x◦y∈Z(R)for all x, y ∈I .

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 82
Then Ris commutative.
Proof. (a) We have [x, y]∈Z(R) for all x, y ∈I. This implies that [r, [x, y]] = 0
for all r∈R. Replace yby yx in the above relation, to get [r, [x, yx]] = [r, [x, y]x] =
[x, y][r, x]+[r, [x, y]]x= 0, now use the relation [r, [x, y]] = 0, to get [x, y][r, x] = 0 for
all x, y ∈I,r∈R. Again, replace rby ry to get [x, y]R(−[x, y]) = {0}for all x, y ∈I
and primeness of Ryields that [x, y] = 0 for all x, y ∈Iand hence by Lemma 2.4, Ris
commutative.
(b) If x◦y∈Z(R) for all x, y ∈I, then [x◦y, r] = 0 for all r∈R. Replacing yby yx we
find that (x◦y)[x, r] = 0. For any s∈R, replace rby sr to get (x◦y)R[x, r] = {0}. Thus,
for each x∈Ieither x◦y= 0 or [x, r] = 0. Let A={x∈I|x◦y= 0 for all y∈I},
B={x∈I|[x, r] = 0 for all r∈R}. Then Aand Bare additive subgroups of Iwhose
union is I. But a group cannot be the union of two proper subgroups and hence either
x◦y= 0 for all x, y ∈Ior [x, r] = 0 for all x∈Iand r∈R. If x◦y= 0, then replace y
by ry we obtain [x, r]y= 0 for all x, y ∈Iand r∈R, that is [x, r]I={0}. Since I6= 0,
we get [x, r] = 0 for all x∈Iand r∈Rand both the cases we find that Iis central and
hence by Remark 2.1, Ris commutative.
3. Main Results
Theorem 3.1. Let Rbe a prime ring and Ia nonzero left ideal of R. Suppose that R
admits a generalized derivation Fwith associated derivation dof Rsuch that d(Z(R)) 6= 0.
Further, if Rsatisfies the condition F(x)◦x∈Z(R)for all x∈I, then Ris commutative.
Proof. By hypothesis we have F(x)◦x∈Z(R) for all x∈I. Replace xby x+y, to
get
F(x)◦y+F(y)◦x∈Z(R) for all x, y ∈I. (1)
Since d(Z(R)) 6= 0, then there exists z∈Z(R) such that d(z)6= 0. Replace yby zy in (1)
and using (1), we get
[F(x), z]y−[z, x]F(y) + d(z)(y◦x)−[d(z), x]y∈Z(R).
Now by Lemma 2.1, d(z)∈Z(R) and therefore we find that d(z)(y◦x)∈Z(R). Since R
is prime and d(z)6= 0, it follows from Remark 2.2 that y◦x∈Z(R) for all x, y ∈Iand
hence by Lemma 2.5(b), Ris a commutative.
Theorem 3.2. Let Rbe a prime ring and Ia nonzero left ideal of Rsuch that I∩Z(R)6= 0.
Suppose that Radmits a generalized derivation Fwith associated derivation dsuch that
d(Z(R)) 6= 0. Further, if Rsatisfies any one of the following conditions:
(i) [F(x), F (y)] −F[x, y]∈Z(R)for al l x, y ∈I, or

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 83
(ii)F(x)◦F(y)−F(x◦y)∈Z(R)for all x, y ∈I ,
then Ris commutative.
Proof. (i) For all x, y ∈I, we have
[F(x), F (y)] −F[x, y]∈Z(R).(2)
Since d(Z(R)) 6= 0, then there exists z∈Z(R) such that d(z)6= 0. Replacing yby zy in
(2) and using (2), we get [F(x), z]F(y)+[F(x), d(z)]y+d(z)([F(x), y]−[x, y]) ∈Z(R) for
all x, y ∈I. Since d(z)∈Z(R) by Lemma 2.1, and therefore d(z)([F(x), y]−[x, y]) ∈Z(R).
Since d(z)6= 0 and Ris prime, it follows from Remark 2.2 that [F(x), y]−[x, y]∈Z(R)
for all x, y ∈I. Now, replace yby d(z)xin the above relation and use it, to get
d(z)[F(x), x]∈Z(R). Again using the same arguments as used above we find that
[F(x), x]∈Z(R) for all x∈I. Thus, by Lemma 2.2 we get Ris commutative.
(ii) For all x, y ∈I, we have
F(x)◦F(y)−F(x◦y)∈Z(R).(3)
Since d(Z(R)) 6= 0, then there exists z∈Z(R) such that d(z)6= 0. Replacing yby zy in (3)
and using (3), we get [F(x), z]F(y) + d(z)(F(x)◦y−x◦y) + [F(x), d(z)]y∈Z(R).Now by
Lemma 2.1 d(z)∈Z(R) and therefore d(z)(F(x)◦y−x◦y)∈Z(R). Since d(z)6= 0 and R
is prim, hence from Remark 2.2 we find that F(x)◦y−x◦y∈Z(R) for all x, y ∈I. Again
replace xby zx in the last expression and use it, to get d(z)(x◦y)−[d(z), y]x∈Z(R).
Now by Lemma 2.1 d(z)∈Z(R) and therefore d(z)(x◦y)∈Z(R) and hence by Remark
2.2, we obtain x◦y∈Z(R) for all x, y ∈I. Thus, by Lemma 2.5 (b), we conclude that R
is commutative.
Theorem 3.3. Let Rbe a prime ring and Ia nonzero left ideal of Rsuch that I∩Z(R)6= 0.
Suppose that Radmits a generalized derivation Fwith associated derivation dsuch that
d(Z(R)) 6= 0. Further, if Rsatisfies the condition:
F[x, y]+[F(x), y]−[F(x), F (y)] ∈Z(R)for all x, y ∈I ,
then Ris commutative.
Proof. For all x, y ∈I, we have
F[x, y]+[F(x), y]−[F(x), F (y)] ∈Z(R).(4)
Since d(Z(R)) 6= 0, then there exists z∈Z(R) such that d(z)6= 0. Replacing yby zy in (4)
and using (4), we get d(z)([x, y]−[F(x), y]) −[F(x), d(z)]y∈Z(R). Since by Lemma 2.1
d(z)∈Z(R) and hence by Remark 2.2 we obtain, [F(x), y] + [x, y]∈Z(R) for all x, y ∈I.
Again replace yby d(z)xin the last relation and use it, to get d(z)[F(x), x]∈Z(R) for
all x∈I, again using the same arguments as above we find that [F(x), x]∈Z(R) for all
x∈Iand hence by Lemma 2.2, Ris commutative.

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 84
Theorem 3.4. Let Rbe a prime ring and Ia nonzero ideal of R. Suppose that Radmits
a generalized derivation Fwith associated derivation dsuch that d(Z(R)) 6= 0. Further,
if Rsatisfies any one of the following conditions:
(i)F(x◦y)−[x, y]∈Z(R)for all x, y ∈I, or
(ii)F(x◦y)+[x, y]∈Z(R)for all x, y ∈I,
then Ris commutative.
Proof. (i) By hypothesis we have F(x◦y)−[x, y]∈Z(R) for all x, y ∈I. If F= 0,
then [x, y]∈Z(R) for all x, y ∈I, and hence by Lemma 2.5(a), we get the required result.
Therefore we shall assume that F6= 0, then we have for any x, y ∈I
F(x◦y)−[x, y]∈Z(R).(5)
Since d(Z(R)) 6= 0, then there exists z∈Z(R) such that d(z)6= 0. Replace yby zy in (5)
to get
zF (x◦y) + d(z)(x◦y)−z[x, y]∈Z(R),for all x, y ∈I,
and hence by (5), we find that d(z)(x◦y)∈Z(R) for all x, y ∈I. Thus, Lemma 2.1 and
Remark 2.2 gives that x◦y∈Z(R) and hence by Lemma 2.5(b) we get the required result.
(ii) Using the same trick as used in (i), result follows.
Theorem 3.5. Let Rbe a prime ring and Ia nonzero left ideal of Rsuch that I∩Z(R)6= 0.
Suppose that Radmits a generalized derivation Fwith associated derivation dsuch that
d(Z(R)) 6= 0. Further, if Rsatisfies any one of the following conditions:
(i) [F(x), F (y)] −x◦y∈Z(R),for all x, y ∈I, or
(ii) [F(x), F (y)] + x◦y∈Z(R),for all x, y ∈I,
then Ris commutative.
Proof. (i) By hypothesis we have [F(x), F (y)] −x◦y∈Z(R) for all x, y ∈I. If F= 0,
then x◦y∈Z(R) for all x, y ∈I, and hence we get the required result by Lemma 2.5(b).
Therefore we shall assume that F6= 0, then for any x, y ∈Iwe have
[F(x), F (y)] −x◦y∈Z(R).(6)
Since d(Z(R)) 6= 0, then there exists z∈Z(R) such that d(z)6= 0. Replacing yby zy
in (6) and using (6), we get d(z)[F(x), y] + [F(x), d(z)]y∈Z(R). Now, since z∈Z(R)
then by Lemma 2.1 we have d(z)[F(x), y]∈Z(R) and hence by Remark 2.2, we find that
[F(x), y]∈Z(R) for all x, y ∈I. In particular [F(x), x]∈Z(R) for all x∈Iand hence by
Lemma 2.2, Ris commutative.
(ii) Using similar arguments as (i) it follows.

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 85
Theorem 3.6. Let Rbe a prime ring and Ia nonzero left ideal of Rsuch that I∩Z(R)6= 0.
Suppose that Radmits a generalized derivations Fand Gwith associated derivations d
and grespectively, such that g(Z(R)) 6= 0. Further, if Rsatisfies any one of the following
conditions:
(i) [F(x), G(y)] −[x, y]∈Z(R),for all x, y ∈I, or
(ii) [F(x), G(y)] + [x, y]∈Z(R),for all x, y ∈I,
then Ris commutative.
Proof. Given that Fand Gare generalized derivations of Rsuch that [F(x), G(y)]−[x, y]∈
Z(R) for all x, y ∈I . If F= 0 (or G= 0), then [x, y]∈Z(R) for all x, y ∈I, and hence
by Lemma 2.5 (a), Ris a commutative.
Therefore, we shall assume that F6= 0 (and G6= 0). For any x, y ∈Iwe have
[F(x), G(y)] −[x, y]∈Z(R).(7)
Since g(Z(R)) 6= 0, then there exists z∈Z(R) such that g(z)6= 0. Replacing yby zy
in (7) and using (7), we get g(z)[F(x), y]+[F(x), g(z)]y∈Z(R) for all x, y ∈I, since by
Lemma 2.1, g(z)∈Z(R) so we find that g(z)[F(x), y]∈Z(R). Thus, by Remark 2.2, we
find that [F(x), y]∈Z(R) for all x, y ∈I. In particular [F(x), x]∈Z(R) for all x∈I.
Hence, Ris commutative by Lemma 2.2.
(ii) Using the same technique as above we get the required result.
Theorem 3.7. Let Rbe a prime ring and Ia nonzero left ideal of Rsuch that I∩Z(R)6= 0.
Suppose that Radmits a generalized derivations Fand Gwith associated derivations d
and grespectively, such that {z∈Z(R)|d(z) = g(z)6= 0} 6=φ. Further, if Rsatisfies
any one of the following conditions:
(i) [F(x), x]−[x, G(x)] ∈Z(R),for all x∈I, or
(ii) [F(x), x]+[x, G(x)] ∈Z(R),for all x∈I,
then Ris commutative.
Proof. (i) It is given that Fand Gare generalized derivations of Rsuch that [F(x), x]−
[x, G(x)] ∈Z(R) for all x∈I. If G= 0 then [F(x), x]∈Z(R) for all x∈I, (or if F= 0,
then −[x, G(x)] ∈Z(R)) and hence in both the cases by Lemma 2.2, we get the required
result.
Henceforth, we shall assume that F6= 0 (and G6= 0). For any x∈I, we have [F(x), x]−
[x, G(x)] ∈Z(R). Linearizing the above expression, we get
[F(x), y]+[F(y), x]−[x, G(y)] −[y, G(x)] ∈Z(R).(8)

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 86
Since {z∈Z(R)|d(z) = g(z)6= 0} 6=φ. Replacing yby zy in (8) and using (8), we find
that d(z)[y, x]+[d(z), x]y−g(z)[x, y]−[x, g(z)]y∈Z(R) for all x, y ∈I. Since z∈Z(R) and
hence by Lemma 2.1, d(z)∈Z(R) and g(z)∈Z(R) and therefore (d(z) + g(z)) ∈Z(R).
Thus, we find that (d(z)+g(z))[y, x]∈Z(R) and hence by Remark 2.2, we get [y, x]∈Z(R)
for all x, y ∈Iand hence by Lemma 2.5 (a), we get the required result.
(ii) Using similar arguments as above it follows.
Theorem 3.8. Let Rbe a prime ring and Ia nonzero ideal of R. Suppose that Radmits
a generalized derivations Fand Gwith associated derivations dand grespectively, such
that {z∈Z(R)|d(z) = g(z)6= 0} 6=φ. Further, if Rsatisfies any one of the following
conditions:
(i)F(x)◦x−x◦G(x)∈Z(R),for all x∈I, or
(ii)F(x)◦x+x◦G(x)∈Z(R),for all x∈I ,
then Ris commutative.
Proof. (i) It is given that Fand Gare generalized derivations of Rsuch that F(x)◦x−x◦
G(x)∈Z(R) for all x∈I. If F= 0, then F(x)◦x∈Z(R) for all x∈I, (or if G= 0, then
−(x◦G(x)) ∈Z(R) and hence in both the cases by Theorem 3.1 we get the required result.
Henceforth, we shall assume that G6= 0 (and F6= 0). For any x∈I, we have F(x)◦x−
x◦G(x)∈Z(R). Linearizing the last expression, to get
F(x)◦y+F(y)◦x−x◦G(y)−y◦G(x)∈Z(R).(9)
Since {z∈Z(R)|d(z) = g(z)6= 0} 6=φ. Replace yby zy in (9) and use (9), to get
d(z)(y◦x)−[d(z), x]y−g(z)(x◦y)−[x, g(z)]y∈Z(R) for all x, y ∈I. Hence, by Lemma
2.1 we find that d(z)∈Z(R) and g(z)∈Z(R) and therefore d(z)−g(z)∈Z(R). Thus,
we obtain (d(z)−g(z))(y◦x)∈Z(R) and by Remark 2.2 it follows that y◦x∈Z(R) for
all x, y ∈Iand hence Ris commutative by Lemma 2.5 (b).
(ii) Using similar arguments as above it follows.
In the next theorem, we consider two identities involving generalized (α, β)−derivation F
associated with (α, β)−derivation d, such that Ris a prime ring with involution ∗, and we
show that Ris commutative.
Theorem 3.9. Let Rbe a 2-torsion free ∗−prime ring and α, β be automorphisms on R.
If Radmits a generalized (α, β)-derivation Fassociated with a nonzero (α, β)−derivation
dsuch that either

M. K. Abu Nawas, R. M. Al-Omary / Eur. J. Pure Appl. Math, 11 (1) (2018), 79-89 87
(i)F[x, y]−[F(x), y]α,β = [d(y), x]α,β for all x, y ∈R, or
(ii)F[x, y]−(F(x)◦y)α,β = [d(y), x]α,β for all x, y ∈R,
then Ris commutative.
Proof. (i) If F= 0, then we have [d(y), x]α,β = 0 for all x, y ∈R. Replacing yby yx in the
last expression gives β(y)[d(x), x]α,β + [β(y), β(x)]d(x) = 0. Again replacing yby zy gives
[β(z), β(x)]β(y)d(x) = 0 for all x, y, z ∈R. But as βis an automorphism on R, we have
[β(z), β(x)]Rd(x) = 0 for all x, z ∈R. (10)
If x∈S∗(R)∩R, then β([z, x])Rd(x) = β([z , x])∗Rd(x).Thus, for some x∈S∗(R)∩R,
the ∗−primeness of Ryields either β([z, x]) = 0 or d(x) = 0. But for any x∈R,
x−x∗∈S∗(R)∩R. Thus, for some x∈R, either [z, x −x∗] = 0 or d(x−x∗) = 0.
If [z, x −x∗] = 0, then equation (10) follows that β([z, x])Rd(x)=0=β([z, x])∗Rd(x).
Hence the ∗−primeness of Ryields either β([z, x]) = 0 or d(x) = 0. If d(x−x∗)=0
then d(x)=(d(x))∗for all x∈R. Consequently, for all x, z ∈R, either β([z, x]) = 0 or
d(x) = 0. Let A={x∈R|d(x)=0}and B={x∈R|[z, x]=0}. Then Aand B
are both additive subgroups of Rwhose union is R. Using Brauer’s trick we have either
A=Ror B=R. If A=Rthen d(x) = 0 for all x∈R, a contradiction. If B=R, then
[z, x] = 0 for all x, z ∈Rand hence Ris commutative.
Therefore, we shall assume that F6= 0. So, for any x, y ∈R, we have
F[x, y]−[F(x), y]α,β = [d(y), x]α,β .(11)
Replacing yby yx in (11) gives
β(y)[x, x]α,β + 2β([x, y])d(x) = β(y)[d(x), x]α,β ,for all x, y ∈R. (12)
Again we replace yby wy in (12) to get 2β([x, w])β(y)d(x) = 0 for all x, y, w ∈R. Since R
is a 2-torsion free and βis an automorphism, we get β([x, w])Rd(x) = {0}, for all x, w ∈R.
Therefore, proceeding in the same way as that after (10), gives the required result.
(ii) If F= 0, then we have [d(y), x]α,β = 0 for all x, y ∈R. Applying the same techniques
as that used above to prove (i) yields the required result. Henceforth, we shall assume
that F6= 0. So, for all x, y ∈R, we have
F[x, y]−(F(x)◦y)α,β = [d(y), x]α,β .(13)
Replacing yby yx in (13) gives
β(y)[x, x]α,β + 2β([x, y])d(x) = β(y)[d(x), x]α,β .(14)
Again, we replace yby wy in (14) to get 2β([x, w])β(y)d(x) = 0 for all x, y ∈R. Since R
is a 2-torsion free and βis an automorphism on R, we get
β([x, w])Rd(x) = {0},for all x, w ∈R. (15)
Now, using similar techniques as that after equation (10), we get the required result.

REFERENCES 88
Acknowledgement
This paper was financially supported by The Deanship of Scientific Research, Northern
Border University, under the project no. 435-062-7. The authors would like to thank The
Deanship of Scientific Research for their financial support. The authors also would like to
thank Professor Nadeem ur Rehman for many useful comments.
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