**Jordan**

*k*

**-Derivations on Lie Ideals of Prime**

## Γ-Rings

**A.C. Paul**

*∗*

### ,

**Ayesha Nazneen**

Department of Mathematics,Rajshahi University, Rajshahi - 6205, Bangladesh
*∗*_{Corresponding Author: acpaulrubd math@yahoo.com}

Copyright © 2014 Horizon Research Publishing All rights reserved.

**Abstract**

Let *M*be a Γ- ring and

*U*a Lie ideal of

*M*. Let

*d*:

*M*

*→*

*M*and

*k*: Γ

*→*Γ be additive mappings. Then

*d*is a

*k*- derivation on

*U*of

*M*if

*d*(*uαv*) =*d*(*u*)*αv*+*uk*(*α*)*v*+*uαd*(*v*) is satisfied for all

*u, v∈U* and*α∈*Γ. And d is a Jordan*k*- derivation on

*U* of*M* if*d*(*uαu*) =*d*(*u*)*αu*+*uk*(*α*)*u*+*uαd*(*u*) holds
for all *u∈U* and*α∈*Γ. It is well-known that every *k*
-derivation on *U* of *M* is a Jordan *k*- derivation on *U*

of *M* but the converse is not true in general. In this
article we prove that every Jordan *k*- derivation on *U*

of*M* is a*k*- derivation on*U* of*M* if ,*M* is a 2- torsion
free prime Γ- ring and*U* is a Lie ideal of*M* such that

*uαu∈U* for all*u∈U* and*α∈*Γ.

**Keywords**

Lie ideal, Jordan *k*- derivation,

*k*-derivation, Prime Γ- ring

**AMS(2010)subject classification:** Primary 16N60,
secondary 03E72,54A40,54B15

**1**

**Introduction**

The notion of a Γ- ring was introduced as an extensive
generalisation of the concept of a classical ring. N.
Nobusawa [10] introduced the notion of a Γ- ring (which
is presently known as a Γ*N*- ring) and afterwords it was

generalised by W.E. Bernes [1] as a more broadsense
(that served us now a days to call it as a Γ- ring
generally). It is wellknown that every ring is a Γ- ring
and also that every Γ*N*- ring is a Γ- ring. We begin

with the following necessary preliminary definitions.
Let M and Γ be additive abelian groups. If there is a
mapping (*x, α, y*)*→* *xαy* of *M* *×*Γ*×M* *→* *M* which
satisfies the conditions

(*a*) (*x* + *y*)*αz* = *xαz* +*yαz, x*(*α* + *β*)*y* = *xαy* +

*xβy, xα*(*y*+*z*) =*xαy*+*xαz* and

(*b*)(*xαy*)*βz*=*xα*(*yβz*) for all*x, y, z∈M* and*α, β∈*Γ,
then*M* is said to be a Γ- ring in the sense of Bernes [1].
In addition to the definition given above, if there is a
mapping (*α, x, β*)*→αxβ* of Γ*×* *M×*Γ*→*Γ satisfying
the conditions

(*a∗*) (*α*+*β*)*xγ* = *αxγ* +*βxγ, α*(*x*+*y*)*β* = *αxβ* +

*αyβ, αx*(*β*+*γ*) =*αxβ*+*αxγ*

(*b∗*) (*xαy*)*βz* = *x*(*αyβ*)*z* = *xα*(*yβz*)

(*c∗*)*xαy* = 0, for all*x, y, z∈M* implies*α*= 0, then*M*

is called a Γ- ring in the sense of nobusawa [12] as simply
a Γ*N*- ring. It is clear that*M* is a Γ*N*- ring implies that

Γ is a*M*- ring.

Let*M* be a Γ- ring . Then *M* is called 2-torsion free if
2*x*= 0 implies*x*= 0 for all*x∈M*. Besides*M* is called
a prime Γ- ring if for all *x, y* *∈M, x*Γ*M*Γ*y* = 0 implies

*x* = 0 or *y* = 0. And *M* is called a semiprime Γ- ring
if for all *x∈M, x*Γ*M*Γ*x*= 0 implies*x*= 0. Note that
every prime Γ- ring is clearly semiprime.

The concept of derivation and Jordan derivation of a
Γ-ring was first introduced by M. Sapanci and A. Nakajima
in [14], whereas the notion of*k*- derivation of a Γ- ring
was used and developed by H. Kandamar [9]. The notion
of Jordan*k*- derivation of a Γ- ring was first initiated by
S. Chakraborty and A.C. Paul [3].

The definition of*k*- derivation and Jordan*k*- derivation
are given as follows :

Let *M* be a Γ- ring. Let *d* : *M* *→* *M* and *k* : Γ *→* Γ
be additive mappings. If *d*(*xαy*) =*d*(*x*)*αy*+*xk*(*α*)*y*+

*xαd*(*y*) is satisfied for all *x, y* *∈* *M* and *α* *∈* Γ, then
d is said to be a *k*- derivation of *M*. And if *d*(*xαx*) =

*d*(*x*)*αx*+*xk*(*α*)*x*+*xαd*(*x*) holds for all*x∈M* and*α∈*Γ,
then *d*is said to be a Jordan*k*- derivation of*M*. Note
that every*k*-derivation is a Jordan*k*- derivation but the
converse is not in general true.

In [2], Y. Ceven proved that every Jordan left derivation
of a 2- torsion free completly prime Γ- ring is a Jordan
left derivation. Paul and Halder [5] extended these
re-sults to a Lie ideal of a Prime Γ- ring. S. Chakraborty
and A.C. Paul [3] worked on a Jordan *k*-derivation and
proved that every Jordan*k*-derivation of a 2- torsion free
Prime Γ*N*- ring is a *k*-derivation.

We shall use the notation [*x, y*]*α* for the commutator *x*

and*y*with respect to*α*, defined by [*x, y*]*α*=*xαy−yαx*.

If*A*is a subset of*M*, by*Z*(*A*) we shall mean the centre
of *A* with respect to *M*, defined by*Z*(*A*) = *{x∈* *M* :
[*x, y*]*α*= 0 for all*a∈A* and*α∈*Γ*}*. The centre of a Γ*−*

ring*M* is denoted by*Z*(*M*) which is defined by*Z*(*M*) =
*{x* *∈* *M* : [*x, y*]*α* = 0 for all*y* *∈* *M* and*α* *∈* Γ*}*. A

Γ-ring*M* is commutative if and only if*M* =*Z*(*M*).
Throughout the paper, we shall use the condition (*)

[*aαb, x*]*β* = [*a, x*]*βαb* + *a*[*α, β*]*xb* + *aα*[*b, x*]*β* and

[*x, aαb*]*β*=*aα*[*x, b*]*β*+*a*[*α, β*]*xb*+[*x, a*]*βαb*given in [7]

re-duce to [*aαb, x*]*β*=*aα*[*b, x*]*β*+ [*a, x*]*βαb*and [*x, aαb*]*β*=
*aα*[*x, b*]*β*+ [*x, a*]*βαb*.

which are used in [7]. F. Hoque and A. C. Paul also used the condition (*) in [8].

In this present article , we introduce the concept of
Jor-dan *k*-derivation on a Lie ideal *U* of a Γ-ring *M*. We
prove that every Jordan*k*-derivation on a Lie ideal*U* of
a 2- torsion free Prime Γ- ring is a*k*-derivation.

**2**

**Lie**

**ideals**

**and**

**Jordan**

*k*

**-derivations**

Let*M* be a Γ-ring. An additive subgroup*U* of*M* is
called a Lie ideal of *M* if [*u, m*]*α* *∈U* for every *u∈* *U*

and *m∈* *M*. Note that every ideal of a Γ-ring *M* is a
Lie ideal of*M* but the converse is not true in general.

**Example 2.1.**

Let R be a commutative ring with
a unity 1 having characteristic 2. Define*M*=

*M*1

*,*2(

*R*)

and Γ =

{(

*n.*1

*n.*1

)

: *n∈*Zand*n*is not divisible by 2

}

.
Then*M* is a Γ- ring . Define *N* =*{*(*a, a*) :*a∈R}*. It
is clear that *N* is an additive subgroup of*M*. Now for

*u*= (*a, a*) *∈N, m*= (*x, y*) *∈M* and *α*=

(

*n*
*n*

)

*∈*Γ
we have ,

*uαm−mαu*= (*a, a*)

(

*n*
*n*

)

(*x, y*)*−*(*x, y*)

(

*n*
*n*

)

(*a, a*)
= (*anx−yna, any−xna*)

= (*anx−*2*yna*+*yna, any−*2*xna*+*xna*)
= (*anx*+*yna, any*+*xna*)

= (*anx*+*any, anx*+*any*)*∈N*

Therefore , *uαm−mαu* *∈* *N* and *N* is a Lie ideal of

*M*. It is clear that*N* is not an ideal of*M*.

In [11] and [12] , Paul and Sabur Uddin worked on Lie and Jordan structure of a 2- torsion free simple Γ- ring and they developed a number of significant results of classical ring thories in Γ- rings.

Now we introduce the concepts of a *k*-derivation , a
Jordan*k*- derivation of Lie ideals in a Γ- ring and then
build up a relationship between these two concepts in a
concrete manner .

Let *M* be a Γ- ring and *U* be a Lie ideal of *M*.
Let *d* : *M* *→* *M* and *k* : Γ *→* Γ be additive
map-pings. If *d*(*uαv*) = *d*(*u*)*αv* +*uk*(*α*)*v* +*uαd*(*v*) is
satisfied for every *u, v* *∈* *U* and *α* *∈* Γ, then *d* is
called a *k* -derivation on a Lie ideal *U* of *M*. And, if

*d*(*uαu*) = *d*(*u*)*αu* +*uk*(*α*)*u*+*uαd*(*u*) holds for all

*u* *∈* *U* and *α* *∈* Γ , then d is said to be a Jordan

*k*-derivation on a Lie ideal*U* of*M*.

It is clear that every *k*-derivation on a Lie ideal is a
Jordan *k*-derivation on a Lie ideal but the converse
may not be true. Now we make an example of a Jordan

*k*- derivation for the case of a Lie ideal which ensures
that Jordan *k*-derivation on a Lie ideal exists and it is
evidently not a*k*- derivation on a Lie ideal.

**Example 2.2.**

Let*M*be a Γ- ring and let

*U*be a Lie ideal of

*M*. Let

*d*:

*M*

*→*

*M*be a

*k*- derivation on a Lie ideal

*U*of

*M*. Define

*M*1 =

*{*(

*x, x*) :

*x∈M}*

and Γ1 = *{*(*α, α*) : *α* *∈* Γ*}*. Define addition and

multiplication on*M*1 as follows :

(*x, x*) + (*y, y*) = (*x*+*y, x*+*y*)*,*

(*x, x*)(*α, α*)(*y, y*) = (*xαy, xαy*).

Under these addition and multiplication, *M*1 is a

Γ1- ring. Define *U*1 = *{*(*u, u*) : *u* *∈* *U}*. Now we

show that *U*1 is a Lie ideal of *M* as follows. For

(*u, u*)*∈U*1*,*(*α, α*)*∈*Γ1 and (*x, x*)*∈M*1, we have,

(*u, u*)(*α, α*)(*x, x*)*−* (*x, x*)(*α, α*)(*u, u*) = (*uαx, uαx*) *−*
(*xαu, xαu*) = (*uαx−xαu, uαx* *−xαu*) *∈* *U*1, since

*uαx−xαu∈U*.

Now let *d*1 : *M*1 *→* *M*1 and *k*1 : Γ1 *→* Γ1 be the

mappings defined by *d*1((*u, u*)) = (*d*(*u*)*, d*(*u*)) for

all *u* *∈* *U,* and*k*1((*α, α*)) = ((*k*(*α*)*, k*(*α*)) for all

*α∈*Γ.Then*d*1and*k*1 are additive mappings. If we say

that (*u, u*) =*u*1*∈U*1for all*u∈U* and (*α, α*) =*γ∈*Γ1

for all*α∈*Γ, then we have

*d*1(*u*1*γu*1) =*d*1((*u, u*)(*α, α*)(*u, u*))

=*d*1((*uαu, uαu*))

= (*d*(*uαu*)*, d*(*uαu*))

= (*d*(*u*)*αu*+*uk*(*α*)*u*+*uαd*(*u*)*, d*(*u*)*αu*+*uk*(*α*)*u*+

*uαd*(*u*))

= (*d*(*u*)*αu, d*(*u*)*αu*) + ((*uk*(*α*)*u, uk*(*α*)*u*) +
(*uαd*(*u*)*, uαd*(*u*)))

= (*d*(*u*)*, d*(*u*))(*α, α*)(*u, u*) + (*u, u*)(*k*(*α*)*, k*(*α*))(*u, u*) +
(*u, u*)(*α, α*)(*d*(*u*)*, d*(*u*))

= *d*1(*u, u*)(*α, α*)(*u, u*) + (*u, u*)*k*1(*α, α*)(*u, u*) +

(*u, u*)(*α, α*)*d*1(*u, u*)

=*d*1(*u*1)*γu*1+*u*1*k*1(*γ*)*u*1+*u*1*γd*1(*u*1).

Hence it follows that *d*1 is a Jordan *k*1- derivation on

a Lie ideal *U*1 of *M*1. It is obvious that *d*1 is not a

*k*1-derivation on a Lie ideal*U* of*M*.

Now we begin with the following results:

**Lemma 2.3**

. Let *M*be a 2- torsion free Γ- ring satisfying the condition (*) and let

*U*be a Lie ideal of

*M* such that *uαu* *∈* *U* for all *u* *∈* *U* and *α* *∈* Γ. Let

*d*:*M* *→M* be a Jordan*k*-derivation on*U* of*M*. Then
for all*u, v, w∈U* and*α, β∈*Γ , we have the following :
(*i*)*d*(*uαv*+*vαu*) =*d*(*u*)*αv*+*uk*(*α*)*v*+*uαd*(*v*) +*d*(*v*)*αu*+

*vk*(*α*)*u*+*vαd*(*u*)

(*ii*)*d*(*uαvβu*) = *d*(*u*)*αvβu*+*uk*(*α*)*vβu*+*uαd*(*v*)*βu*+

*uαvk*(*β*)*u*+*uαvβd*(*u*)

(*iii*)*d*(*uαvβw*+*wαvβu*) =*d*(*u*)*alphavβw*+*uk*(*α*)*vβw*+

*uαd*(*v*)*βw* + *uαvk*(*β*)*w* + *uαvβd*(*w*) + *d*(*w*)*αvβu* +

*wk*(*α*)*vβu*+*wαd*(*v*)*βu*+*wαvk*(*β*)*u*+*wαvβd*(*u*).

**Proof.**

(i) Since *uαv*+*vαu* = (*u*+*v*)*α*(*u*+*v*)*−uαu−vαv*

and the right side is in *U*, we have the left side of the
identity is in*U*. Hence

*d*(*uαv*+*vαu*) =*d*((*u*+*v*)*α*(*u*+*v*)*−uαu−vαv*)
=*d*(*u*+*v*)*α*(*u*+*v*)+(*u*+*v*)*k*(*α*)(*u*+*v*)+(*u*+*v*)*αd*(*u*+*v*)*−*
(*d*(*u*)*αu*+*uk*(*α*)*u*+*uαd*(*u*) +*d*(*v*)*αv*+*vk*(*α*)*v*+*vαd*(*v*))
= (*d*(*u*) +*d*(*v*))*α*(*u*+*v*) + (*u*+*v*)*k*(*α*)(*u*+*v*) + (*u*+

*v*)*α*(*d*(*u*) +*d*(*v*))*−d*(*u*)*αu−uk*()*u−uαd*(*u*)*−d*(*v*)*αv−*
*vk*(*α*)*v−vαd*(*v*)

=*d*(*u*)*αu*+*d*(*u*)*αv*+*d*(*v*)*αu*+*d*(*v*)*αv*+*uk*(*α*)*u*+*uk*(*α*)*v*+

=*d*(*u*)*αv*+*uk*(*α*)*v*+*uαd*(*v*)+*d*(*v*)*αu*+*vk*(*α*)*u*+*vαd*(*u*)*.*

(*ii*) Replace*v* by*uβv*+*vβu*in (*i*) we have

*d*(*uα*(*uβv*+*vβu*) + (*uβv*+*vβu*)*αu*)

=*d*(*u*)*α*(*uβv*+*vβu*) +*uk*(*α*)(*uβv*+*vβu*) +*uαd*(*uβv*+

*vβu*) +*d*(*uβv*+*vβu*)*αu*+ (*uβv*+*vβu*)*k*(*α*)*u*+ (*uβv*+

*vβu*)*αd*(*u*)

=*⇒d*(*uαuβv*+*uαvβu*+*uβvαu*+*vβuαu*) =*d*(*u*)*α*(*uβv*+

*vβu*)+*uk*(*α*)(*uβv*+*vβu*)+*uα*(*d*(*u*)*βv*+*uk*(*β*)*v*+*uβd*(*v*)+

*d*(*v*)*βu*+*vk*(*β*)*u*+*vβd*(*u*) + (*d*(*u*)*βv*+*uk*(*β*)*v*+*uβd*(*v*) +

*d*(*v*)*βu*+*vk*(*β*)*u*+*vβd*(*u*))*αu*+(*uβv*+*vβu*)*k*(*α*)*u*+(*uβv*+

*vβu*)*αd*(*u*)
Here,

*d*((*uαu*)*βv* +*vβ*(*uαu*)) = *d*(*uαu*)*βv* + (*uαu*)*k*(*β*)*v* +
(*uαu*)*βd*(*v*) +*d*(*v*)*β*(*uαu*) +*vk*(*β*)(*uαu*) +*vβd*(*uαu*)
= *d*(*u*)*αuβv* +*uk*(*α*)*uβv* +*uαd*(*u*)*βv* + *uαuk*(*β*)*v* +

*uαuβd*(*v*) + *d*(*v*)*βuαu* + *vk*(*β*)*uαu* + *vβd*(*u*)*αu* +

*vβuk*(*α*)*u*+*vβuαd*(*u*).
Therefore we have,

*d*(*uαvβu*+*uβvαu*)+*d*(*u*)*αuβv*+*uk*(*α*)*uβv*+*uαd*(*u*)*βv*+

*uαuk*(*β*)*v* + *uαuβd*(*v*) + *d*(*v*)*βuαu* + *vk*(*β*)*uαu* +

*vβd*(*u*)*αu* + *vβuk*(*α*)*u* + *vβuαd*(*u*) = *d*(*u*)*αuβv* +

*d*(*u*)*αvβu* + *uk*(*α*)*uβv* + *uk*(*α*)*vβu* + *uαd*(*u*)*βv* +

*uαuk*(*β*)*v* + *uαuβd*(*v*) + *uαd*(*v*)*βu* + *uαvk*(*β*)*u* +

*uαvβd*(*u*) + *d*(*u*)*βvαu* + *uk*(*β*)*vαu* + *uβd*(*v*)*αu* +

*d*(*v*)*βuαu* + *vk*(*β*)*uαu* + *vβd*(*u*)*αu* + *uβvk*(*α*)*u* +

*vβuk*(*α*)*u*+*uβvαd*(*u*) +*vβuαd*(*u*)

=*⇒* *d*(*uαvβu*+*uβvαu*) = *d*(*u*)*αvβu* +*uk*(*α*)*vβu*+

*uαd*(*v*)*βu* + *uαvk*(*β*)*u* + *uαvβd*(*u*) + *d*(*u*)*βvαu* +

*uk*(*β*)*vαu*+*uβd*(*v*)*αu*+*uβvk*(*α*)*u*+*uβvαd*(*u*).
Put*uαvβu*=*uβvαu*, we have

*d*(2*uαvβu*) = *d*(*u*)*αvβu* + *uk*(*α*)*vβu* + *uαd*(*v*)*βu* +

*uαvk*(*β*)*u* + *uαvβd*(*u*) + *d*(*u*)*αvβu* + *uαvk*(*β*)*u* +

*uαd*(*v*)*βu*+*uk*(*α*)*vβu*+*uαvβd*(*u*)

=*⇒*2*d*(*uαvβu*) = 2(*d*(*u*)*αvβu*+*uk*(*α*)*vβu*+*uαd*(*v*)*βu*+

*uαvk*(*β*)*u*+*uαvβd*(*u*))

Since*M* is 2- torsion free, hence

*d*(*uαvβu*) = *d*(*u*)*αvβu* + *uk*(*α*)*vβu* + *uαd*(*v*)*βu* +

*uαvk*(*β*)*u*+*uαvβd*(*u*)*.*

(*iii*) Replace*u*+*w*for u in (ii) we have ,

*d*((*u* + *w*)*αvβ*(*u* + *w*)) = *d*(*u* + *w*)*αvβ*(*u* + *w*) +
(*u*+*w*)*k*(*α*)*vβ*(*u*+*w*) + (*u*+*w*)*αd*(*v*)*β*(*u*+*w*) + (*u*+

*w*)*αvk*(*β*)(*u*+*w*) + (*u*+*w*)*αvβd*(*u*+*w*)*.*

=*⇒d*(*uαvβu*+*uαvβw*+*wαvβu*+*wαvβw*) = (*d*(*u*) +

*d*(*w*))*αvβ*(*u* + *w*) + (*u* + *w*)*k*(*α*)*vβ*(*u* + *w*) + (*u* +

*w*)*αd*(*v*)*β*(*u* + *w*) + (*u* + *w*)*αvk*(*β*)(*u* + *w*) + (*u* +

*w*)*αvβ*(*d*(*u*) +*d*(*w*))
Here we have

*d*(*uαvβu*+*wαvβw*) =*d*(*uαvβu*) +*d*(*wαvβw*)

= *d*(*u*)*αvβu* +*uk*(*α*)*vβu* +*uαd*(*v*)*βu*+ *uαvk*(*β*)*u*+

*uαvβd*(*u*) + *d*(*w*)*αvβw* + *wk*(*α*)*vβw* + *wαd*(*v*)*βw* +

*wαvk*(*β*)*w*+*wαvβd*(*w*).
And also

*d*(*uαvβu*+*wαvβu*+*uαvβw*+*wαvβw*) = *d*(*uαvβw*+

*wαvβu*) +*d*(*uαvβu*+*wαvβw*).
Hence we have

*d*(*uαvβw*+*wαvβu*)+*d*(*u*)*αvβu*+*uk*(*α*)*vβu*+*uαd*(*v*)*βu*+

*uαvk*(*β*)*u* + *uαvβd*(*u*) + *d*(*w*)*αvβw* + *wk*(*α*)*vβw* +

*wαd*(*v*)*βw* +*wαvk*(*β*)*w*+ *wαvβd*(*w*) = *d*(*u*)*αvβu* +

*d*(*w*)*αvβu* + *d*(*u*)*αvβw* + *d*(*w*)*αvβw* + *uk*(*α*)*vβu* +

*wk*(*α*)*vβu* + *uk*(*α*)*vβw* + *wk*(*α*)*vβw* + *uαd*(*v*)*βu* +

*wαd*(*v*)*βu* + *uαd*(*v*)*βw* + *wαd*(*v*)*βw* + *uαvk*(*β*)*u* +

*wαvk*(*β*)*u* + *uαvk*(*β*)*w* + *wαvk*(*β*)*w* + *uαvβd*(*u*) +

*wαvβd*(*u*) +*uαvβd*(*w*) +*wαvβd*(*w*).

=*⇒* *d*(*uαvβw*+*wαvβu*) = *d*(*u*)*αvβw* +*uk*(*α*)*vβw*+

*uαd*(*v*)*βw* + *uαvk*(*β*)*w* + *uαvβd*(*w*) + *d*(*w*)*αvβu* +

*wk*(*α*)*vβu*+*wαd*(*v*)*βu*+*wαvk*(*β*)*u*+*wαvβd*(*u*)*.*

**Definition 2.4**

We define*ϕα*(*u, v*) =*d*(*uαv*)*−d*(*u*)*αv−uk*(*α*)*v−uαd*(*v*)

for every *u, v∈U* and*α∈*Γ*.*

**Remark**

It is clear that*d*is a*k*-derivation on*U* of*M* if and only
if*ϕα*(*u, v*) = 0*.*

**Lemma 2.5**

. Let *M, U*and

*d*be as in Lemma 2.3.Then for all

*u, v, w∈U*and

*α, β∈*Γ, the following relations hold .

(*i*)*ϕα*(*u, v*) +*ϕα*(*v, u*) = 0

(*ii*)*ϕα*(*u*+*w, v*) =*ϕα*(*u, v*) +*ϕα*(*w, v*)

(*iii*)*ϕα*(*u, v*+*w*) =*ϕα*(*u, v*) +*ϕα*(*u, w*)

(*iv*)*ϕα*+*β*(*u, v*) =*ϕα*(*u, v*) +*ϕβ*(*u, v*)

**Lemma 2.6**

. Let *M, U*and

*d*be as in Lemma 2.3, then for all

*u, v, w∈U*and

*α, β, γ,∈*Γ,

*ϕα*(*u, v*)*βwγ*[*u, v*]*α*+ [*u, v*]*αβwγϕα*(*u, v*) = 0*.*

**Proof.**

Consider *A*= (2*uαv*)*βwγ*(2*vαu*) + (2*vαu*)*βwγ*(2*uαv*)
=*⇒d*(*A*) =*d*((2*uαv*)*βwγ*(2*vαu*) + (2*vαu*)*βwγ*(2*uαv*))
= *d*(2*uαv*)*βwγ*(2*vαu*) + 2*uαvk*(*β*)*wγ*(2*vαu*) +
(2*uαv*)*βd*(*w*)*γ*(2*vαu*) + (2*uαv*)*βwk*(*γ*)(2*vαu*) +
(2*uαv*)*βwγd*(2*vαu*) + *d*(2*vαu*)*βwγ*(2*uαv*) +
(2*vαu*)*k*(*β*)*wγ*(2*uαv*) + (2*vαu*)*βd*(*w*)*γ*(2*uαv*) +
(2*vαu*)*βwk*(*γ*)(2*uαv*) + (2*vαu*)*βwγd*(2*uαv*)

= 4*d*(*uαv*)*βwγ*(*vαu*) + 4(*uαv*)*k*(*β*)*wγ*(*vαu*) +
4(*uαv*)*βd*(*w*)*γ*(*vαu*) + 4(*uαv*)*βwk*(*γ*)(*vαu*) +
4(*uαv*)*βwγd*(*vαu*) + 4*d*(*vαu*)*βwγ*(*uαv*) +
4(*vαu*)*k*(*β*)*wγ*(*uαv*) + 4(*vαu*)*βd*(*w*)*γ*(*uαv*) +
4(*vαu*)*βwk*(*γ*)(*uαv*) + 4(*vα*)*βwγd*(*uαv*)

Again*A*= (2*uαv*)*βwγ*(2*vαu*) + (2*vαu*)*βwγ*(2*uαv*)
=*uα*(4*vβwγv*)*αu*+*vα*(4*uβwγu*)*αv*

=*⇒d*(*A*) =*d*(*uα*(4*vβwγv*)*αu*) +*d*(*vα*(4*uβwγu*)*αv*)
= *d*(*u*)*α*(4*vβwγv*)*αu* + *uk*(*α*)(4*vβwγv*)*αu* +

*uαd*(4*vβwγv*)*αu* + *uα*(4*vβwγv*)*k*(*α*)*u* +

*uα*(4*vβwγv*)*αd*(*u*) + *d*(*v*)*α*(4*uβwγu*)*αv* +

*vk*(*α*)(4*uβwγu*)*αv* + *vαd*(4*uβwγu*)*αv* +

*vα*(4*uβwγu*)*k*(*α*)*v*+*vα*(4*uβwγu*)*αd*(*v*)

= 4*d*(*u*)*αvβwγvαu*+4*uk*(*α*)*vβwγvαu*+4*uα*(*d*(*v*)*βwγv*+

*vk*(*β*)*wγv* +*vβd*(*w*)*γv* +*vβwk*(*γ*)*v* +*vβwγd*(*v*))*αu* +
4*uαvβwγvk*(*α*)*u*+4*uαvβwγvαd*(*u*)+4*d*(*v*)*αuβwγuαv*+
4*vk*(*α*)*uβwγuαv* + 4*vα*(*d*(*u*)*βwγu* + *uk*(*β*)*wγu* +

*uβd*(*w*)*γu* + *uβwk*(*γ*)*u* + *uβwγd*(*u*))*αv* +
4*vαuβwγuk*(*α*)*v*+ 4*vαuβwγuαd*(*v*)

= 4*d*(*u*)*αvβwγvαu* + 4*uk*(*α*)*vβwγvαu* +
4*uαd*(*v*)*βwγvαu*+4*uαvk*(*β*)*wγvαu*+4*uαvβd*(*w*)*γvαu*+
4*uαvβwk*(*γ*)*vαu*+4*uαvβwγd*(*v*)*αu*+4*uαvβwγvk*(*α*)*u*+
4*uαvβwγvαd*(*u*)+4*d*(*v*)*αuβwγuαv*+4*vk*(*α*)*uβwγuαv*+
4*vαd*(*u*)*βwγuαv*+4*vαuk*(*β*)*wγuαv*+4*vαuβd*(*w*)*γuαv*+
4*vαuβwk*(*γ*)*uαv* + 4*vαuβwγd*(*u*))*αv* +
4*vαuβwγuk*(*α*)*v*+ 4*vαuβwγuαd*(*v*)

4*uαvβwγ*(*d*(*vαu* *−* *d*(*v*)*αu* *−* *vk*(*α*)*u* *−* *vαd*(*u*)) +
4(*d*(*vαu*) *−* *d*(*v*)*αu* *−* *vk*(*α*)*u* *−* *vαd*(*u*))*βwγuαv* +
4*vαuβwγ*(*d*(*uαv*)*−d*(*u*)*αv−uk*(*α*)*v−uαd*(*v*)) = 0
=*⇒* 4(*ϕα*(*u, v*)*βwγvαu* *−* *ϕα*(*u, v*)*βwγuαv* *−*
*uαvβwγϕα*(*v, u*) +*vαuβwγϕα*(*u, v*)) = 0

Since*M* is 2- torsion free, we have

*−ϕα*(*u, v*)*βwγ*(*uαv−vαu*)*−*(*uαv−vαu*)*βwγϕα*(*u, v*) =
0

*⇒ϕα*(*u, v*)*βwγ*[*u, v*]*α*+ [*u, v*]*αβwγϕα*(*u, v*) = 0*.*

**Lemma 2.7.**

Let *U*

*̸⊆*

*Z*(

*M*) be a Lie ideal of a 2-torsion free prime Γ-ring

*M*, then

*Z*(

*U*) =

*Z*(

*M*)

*.*

**Proof.**

We have*Z*(

*U*) is both a sub Γ

*−*ring and a Lie ideal of

*M*. Also we know that

*Z*(

*U*)cannot contain a nonzero ideal of

*M*. So by [9, Lemma 3.7],

*Z*(

*U*) is contained in

*Z*(

*M*).Therefore,

*Z*(

*U*) =

*Z*(

*M*).

**Lemma 2.8.**

Let *U*be a Lie ideal of a 2-torsion free prime Γ-ring

*M*satisfying the condition (*) and

*a* *∈* *M*. If *a* *∈* *Z*([*U, U*]Γ), then *a* *∈* *Z*(*U*). That is

*Z*([*U, U*]Γ) =*Z*(*U*).

**Proof.**

Obviously *Z*(

*u*)

*⊆Z*([

*U, U*]Γ).

If *Z*([*U, U*]Γ) *̸⊆* *Z*(*M*), then by Lemma 2.7, *a* *∈*

*Z*(*M*)*⇒a∈Z*(*U*)

On the other hand if *Z*([*U, U*]Γ) *⊆* *Z*(*M*), then for all

*u* *∈* *U, m* *∈* *M, α, β* *∈* Γ implies *a* = [*u,*[*u, m*]*α*]*β* *∈*
*Z*(*M*).

Using the condition (*) we have

*aγu*= [*u,*[*u, uγm*]*α*]*β* *∈Z*(*M*).

If *a̸*=*o*, we get *u∈Z*(*M*) implies*a*= 0
Thus [*u,*[*u, uγm*]*α*]*β* = 0 for all*m∈M*.

By the subLemma 3.8 of [9] *u* *∈* *Z*(*M*); hence *U* *⊆*
*Z*(*M*).

In both cases we see that *a* *∈* *Z*(*U*). This gives that

*Z*([*U, U*]Γ) =*Z*(*U*)*.*

**Lemma 2.9.**

Let *U*

*̸⊆Z*(

*M*) be a Lie ideal of a 2-torsion free Γ-ring

*M*satisfying the condition (*) such that ,

*uαu∈*

*U*for all

*u∈U*and

*α∈*Γ. If

*u∈*

*Z*(

*U*) then

*d*(

*u*)

*∈Z*(

*M*).

**Proof.**

Let *u*

*∈*

*Z*(

*U*) =

*Z*(

*M*), then ,

*uαv*=

*vαu,* for every *v∈U* and*α∈*Γ*.*

From Lemma 2.3(i) we have,

*d*(*uαv*+*vαu*) =*d*(*u*)*αv*+*uk*(*α*)*v*+*uαd*(*v*) +*d*(*v*)*αu*+

*vk*(*α*)*u*+*vαd*(*u*)

=*⇒* *d*(2*uαv*) = *d*(*u*)*αv*+*vαd*(*u*) +*uk*(*α*)*v*+*uαd*(*v*) +

*uαd*(*v*) +*uk*(*α*)*v*

=*d*(*u*)*αv*+*vαd*(*u*) + 2*uk*(*α*)*v*+ 2*uαd*(*v*)
Replace*v*by (*vβw*+*wβv*),we have

*d*(2*uα*(*vβw*+*wβv*)) = *d*(*u*)*α*(*vβw* +*wβv*) + (*vβw*+

*wβv*)*αd*(*u*) + 2*uk*(*α*)(*vβw*+*wβv*) + 2*uαd*(*vβw*+*wβv*)
=*⇒* *d*(*uαvβw*+*uαwβv*) = *d*(*u*)*αvβw* +*d*(*u*)*αwβv* +

*vβwαd*(*u*) +*wβvαd*(*u*) + 2*uk*(*α*)*vβw* + 2*uk*(*α*)*wβv* +
2*uαd*(*v*)*βw*+ 2*uαvk*(*β*)*w*+ 2*uαvβd*(*w*) + 2*uαd*(*w*)*βv*+
2*uαwk*(*β*)*v*+ 2*uαwβd*(*v*)

Now (2*uαvβw*+ 2*uαwβv*) = 2*d*(*uαvβw*+*wαvβu*)
= 2*d*(*u*)*αvβw*+2*uk*(*α*)*vβw*+2*uαd*(*v*)*βw*+2*uαvk*(*β*)*w*+
2*uαvβd*(*w*) + 2*d*(*w*)*αvβu*+ 2*wk*(*α*)*vβu*+ 2*wαd*(*v*)*βu*+
2*wαvk*(*β*)*u*+ 2*wαvβd*(*u*)

So 2*d*(*u*)*αvβw*+2*uk*(*α*)*vβw*+2*uαd*(*v*)*βw*+2*uαvk*(*β*)*w*+
2*uαvβd*(*w*) + 2*d*(*w*)*αvβu*+ 2*wk*(*α*)*vβu*+ 2*wαd*(*v*)*βu*+
2*wαvk*(*β*)*u*+ 2*wαvβd*(*u*) = *d*(*u*)*αvβw*+*d*(*u*)*αwβv* +

*vβwαd*(*u*) +*wβvαd*(*u*) + 2*uk*(*α*)*vβw*+ 2*uk*(*α*)*wβv*+
2*uαd*(*v*)*βw*+ 2*uαvk*(*β*)*w*+ 2*uαvβd*(*w*) + 2*uαd*(*w*)*βv*+
2*uαwk*(*β*)*u*+ 2*uαwβd*(*v*)

=*⇒d*(*u*)*αvβw*+2*d*(*w*)*αvβu*+2*wk*(*α*)*vβu*+2*wαd*(*v*)*βu*+
2*wαvk*(*β*)*u*+ 2*wβvαd*(*u*) = *d*(*u*)*αwβv* +*vβwαd*(*u*) +

*wβvαd*(*u*) + 2*wk*(*α*)*vβu*+ 2*d*(*w*)*αvβu*+ 2*wαvk*(*β*)*u*+
2*wαd*(*v*)*βu,*

=*⇒d*(*u*)*vβw*+ 2*d*(*w*)*αvβu*+ 2*wk*(*α*)*vβu*+ 2*wαd*(*v*)*βu*+
2*wαvk*(*β*)*u*+ 2*wαvβd*(*u*) = *d*(*u*)*αwβv* +*vβwαd*(*u*) +

*wβvαd*(*u*) + 2*uk*(*α*)*wβv*+ 2*uαd*(*w*)*βv*+ 2*uαwk*(*β*)*v*+
2*uαwβd*(*v*)

=*⇒d*(*u*)*α*(*vβw−wβv*) = (*vβw−wβv*)*αd*(*u*)
=*⇒d*(*u*)*∈Z*([*U, U*]Γ)

But by Lemma 2.7 and Lemma 2.8 , we have,

*Z*([*U, U*]Γ) =*Z*(*M*)*.*Hence *d*(*u*)*∈Z*(*M*)*.*

To prove our main results we need the following two Lemmas .

**Lemma 2.10 [ 13, Lemma 2.10]**

Let *U*be a Lie ideal of a 2- torsion free Prime Γ- ring satisfying the condition (*) and

*U*

*̸⊆Z*(

*M*). If

*a, b∈*

*M*(res.

*b*

*∈U*

and*a∈M* ) such that*aαU βb*= 0 for all*α, β∈*Γ, then

*a*= 0 or*b*= 0.

**Lemma 2.11**

**[13, Lemma 2.11 ]**

Let
*U* *̸⊆* *Z*(*M*) be a 2- torsion free lie ideal of a prime
Γ-ring*M*. If*a, b∈M* (res.*a∈M* and*b∈U*) such that

*aαxβb*+*bαxβa*= 0 for all*x∈* *U* and *α, β* *∈*Γ, then

*aαxβb*=*bαxβa*= 0.

Now we have in position to prove our main result.

**Theorem 2.12**

. Let *M*be a 2 -torsion free prime Γ-ring satisfying the condition (*) and let

*U*be a Lie ideal of

*M*such that

*uαu∈U*for all

*u∈U*and

*α,∈*Γ. If

*d*:

*M*

*−→*

*M*is a Jordan

*k*-derivation on

*U*of

*M*, then

*d*is a

*k*-derivation on

*U*of

*M*.

**Proof.**

If*U*is a commutative Lie ideal of

*M*, then for all

*u, v*

*∈U*and

*α∈*Γ

*,*[

*u, v*]

*α*= 0. Then

*uαv*=

*vαu.*

By Lemma 2.3(iii), we have

*d*(*uαvβw* + *wαvβu*) = *d*(*u*)*αvβw* + *uk*(*α*)*vβw* +

*uαd*(*v*)*βw* + *uαvk*(*β*)*w* + *uαvβd*(*w*) + *d*(*w*)*αvβu* +

*wk*(*α*)*vβu*+*wαd*(*v*)*βu*+*wαvk*(*β*)*u*+*wαvβd*(*u*)*.*

By using (*) we obtain,

*d*(*uαvβw*+*wαvβu*) =*d*((*uαv*)*βw*+*wβ*(*uαv*))

=*d*(*uαv*)*βw*+(*uαv*)*k*(*β*)*w*+(*uαv*)*βd*(*w*)+*d*(*w*)*β*(*uαv*)+

*wk*(*β*)(*uαv*) +*wβd*(*uαv*).

Comparing the above two expressions, we obtain

*d*(*uαv*)*βw*+ (*uαv*)*k*(*β*)*w*+ (*uαv*)*βd*(*w*) +*d*(*w*)*β*(*uαv*) +

*wk*(*β*)(*uαv*) +*wβd*(*uαv*) = *d*(*u*)*αvβw* +*uk*(*α*)*vβw*+

*uαd*(*v*)*βw* + (*uαv*)*k*(*β*)*w* + *uαvβd*(*w*) + *d*(*w*)*βuαv* +

*wk*(*β*)*uαv*+*wβd*(*v*)*αu*+*wβvk*(*α*)*u*+*wβvαd*(*u*)
*⇒*(*d*(*uαv*)*−d*(*u*)*αv−uk*(*α*)*v−uαd*(*v*))*βw*+*wβ*(*d*(*vαu*)*−*

*d*(*v*)*αu−vk*(*α*)*u−vαd*(*u*)) = 0
*⇒ϕα*(*u, v*)*βw*+*wβϕα*(*v, u*) = 0
*⇒ϕα*(*u, v*)*βw−wβϕα*(*u, v*) = 0

*⇒ϕα*(*u, v*)*βw*=*wβϕα*(*u, v*), for all*w∈U, β∈*Γ*.*

Then*ϕα*(*u, v*)*∈Z*(*U*) =*Z*(*M*) by Lemma 2.7.

since*uαu∈U* and(*uαu*)*βv*=*vβ*(*uαu*) for all *β∈*Γ
. Hence, *d*(*uαuβv*) *−* *d*(*uαu*)*βv* *−* (*uαu*)*k*(*β*)*v* *−*

(*uαu*)*βd*(*v*)*∈Z*(*M*)

*⇒* *d*(*uαuβv*) *−* ((*d*(*u*)*αu* + *uk*(*α*)*u* + *uαd*(*u*))*βv* +
(*uαu*)*k*(*β*)*v*+*uαuβd*(*v*))*∈Z*(*M*))...(1)

Also, 2*uβv∈U* and*uα*(2*uβv*) = (2*uβv*)*αu*, we get

*d*(*uα*(2*uβv*)*−d*(*u*)*αuβv*)*−uk*(*α*)(2*uβv*)*−uαd*(2*uβv*)*∈*

*Z*(*M*)

i.e,2(*d*(*uαuβv*)*−d*(*u*)*αuβv−uk*(*α*)*uβv−uαd*(*uβv*))*∈*

*Z*(*M*)*.*

Since*M* is a 2-torsion free , we have

*d*(*uαuβv*) *−* *d*(*u*)*αuβv* *−* *uk*(*α*)*uβv* *−* *uαd*(*uβv*) *∈*

*Z*(*M*)*...*(2)

From (1) and (2) we have *d*(*uαuβv*) *−d*(*u*)*αuβv* *−*
*uk*(*α*)*uβv* *−* *uαd*(*u*)*βv* *−* *uαuk*(*β*)*v* *−* *uαuβd*(*v*) *−*

*d*(*uαuβv*) +*d*(*u*)*αuβv*+*uk*(*α*)*uβv*+*uαd*(*uβv*)
=*uαd*(*uβv*)*−uαuk*(*β*)*v−uαuβd*(*v*)*−uαd*(*u*)*βv*

=*uα*(*d*(*uβv*)*−d*(*u*)*βv−uk*(*β*)*v−uβd*(*v*))
=*uαϕβ*(*u, v*)*∈Z*(*M*)*.*

If*ϕβ*(*u, v*)*̸*= 0. Since*M* is prime and*ϕβ*(*u, v*)*∈Z*(*M*),
then*u∈Z*(*M*)*.*

So*d*(*u*)*∈Z*(*M*)

By Lemma 2.3(i), we have,

*d*(*uαv*+*vαu*) =*d*(*u*)*αv*+*uk*(*α*)*v*+*uαd*(*v*) +*d*(*v*)*αu*+

*vk*(*α*)*u*+*vαd*(*u*)

*⇒d*(2*uαv*) = 2(*d*(*u*)*αv*+*uk*(*α*)*v*+*uαd*(*v*))
*⇒*2(*d*(*uαv*)*−d*(*u*)*αv−uk*(*α*)*v−uαd*(*v*)) = 0
*⇒*2*ϕα*(*u, v*) = 0*⇒ϕα*(*u, v*) = 0

Again, let*U* is not commutative. i.e,*U* *̸⊆Z*(*M*) . Then
by lemma 2.6 , we have

(*i*)*...ϕα*(*u, v*)*βwγ*[*u, v*]*α*+ [*u, v*]*αβwγϕα*(*u, v*) = 0.

Applying Lemma 2.11 in (i) , we obtain
(*ii*)*...ϕα*(*u, v*)*βwγ*[*u, v*]*α*= 0 and

(*iii*)*...*[*u, v*]*αβwγϕα*(*u, v*) = 0.

In view of Lemma 2.10, we have from (ii) that*ϕα*(*u, v*) =
0 or [*u, v*]*α*= 0.

The same result follows from (iii) by applying Lemma 2.10 .

For every*v∈U*, let us define

*A*=*{u∈U* :*ϕα*(*u, v*) = 0*}* and
*B*=*{u∈U* : [*u, v*]*α*= 0*}*.

Then *A* and *B* are additive subgroup of *U* such that

*A∪B*=*U*, Therefore , by Brauer’s trick, either*A*=*U*

or *B* = *U*. By using the same argument,we have

*U* = *{v* *∈* *U* : *U* = *A}* and *U* = *{v* *∈* *U* : *U* = *B}*

For the later case , we have*U* *⊆Z*(*M*) which is a
con-tradiction. So, we have *ϕα*(*u, v*) = 0*,* which completes
the proof.

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