# Jordan Γ* Derivation on Semiprime Γ Ring M with Involution

11

## Full text

(1)

http://dx.doi.org/10.4236/alamt.2016.62006

1

1

### , Abdul-Rahman Hameed Majeed

2

1School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia 2Department of Mathematics, University of Baghdad, Baghdad, Iraq

Received 21 March 2016; accepted 30 May 2016; published 2 June 2016

### Abstract

Let M be a 2-torsion free semiprime Γ-ring with involution satisfying the condition that

(a b c, ,M and

### ∈ Γ

). In this paper, we will prove that if a non-zero Jordan

Γ*-derivation d on M satisfies

### ( )

d x ,xαZ M for all xM and

∈ Γ, then d x

,xα =0.

### Keywords

Γ-Ring M with Involution, Jordan Γ*-Derivation, Commutative Γ-Ring

### 1. Introduction

The notion of Γ-ring was introduced as a generalized extension of the concept on classical ring. From its first appearance, the extensions and the generalizations of various important results in the theory of classical rings to the theory of Γ-rings have attracted a wider attention as an emerging field of research to enrich the world of al-gebra. A good number of prominent mathematicians have worked on this interesting area of research to develop many basic characterizations of Γ-rings. Nobusawa [1] first introduced the notion of a Γ-ring and showed that

Γ-rings are more general than rings. Barnes [2] slightly weakened the conditions in the definition of Γ-ring in the sense of Nobusawa. Barnes [2], Luh [3], Kyuno [4], Hoque and Pual [5]-[7], Ceven [8], Dey et al. [9][10], Vukman [11] and others obtained a large number of important basic properties of Γ-rings in various ways and developed more remarkable results of Γ-rings. We start with the following necessary introductory definitions.

Let M and Γ be additive abelian groups. If there exists an additive mapping

x, ,α y

x yα

### )

of

M× Γ×MM which satisfies the conditions: 1)

,

(2)

3)

x yα β

z=xα β

y z

### )

, then M is called a Γ-ring [2]. Every ring M is a Γ-ring with M = Γ. However a Γ- ring need not be a ring. Let M be a Γ-ring. Then M is said to be prime if a M bΓ Γ =

0 with

### M

, implies

0

a= or b=0 and semiprime if a M aΓ Γ =

### ( )

0 with aM implies a=0. Furthermore, M is said to be a commutative Γ-ring if x y

=y x

for all

and

### α

∈ Γ. Moreover, the set

: for all ,

### }

Z M = xM x yα = y xα α∈ Γ ∈y M is called the center of the Γ-ring M. If M is a Γ-ring, then

x y, α =x y

y x

### α

is known as the commutator of x and y with respect to

, where

and

### α

∈ Γ. We make the following basic commutator identities:

,

,

,

### [ ]

,

z

x y zα β = x z βαy+x α β y+xα y z β (1)

x y z, α

β = x y, βαz+y

α β,

xz+yα

### [ ]

x z, β (2) for all

and

### ,

. Now, we consider the following assumption:

A...

, for all

and

### ,

.

According to assumption (A), the above commutator identities reduce to

x y zα ,

β = x z, βαy+xα

y z, β and

x y z, α

β = x y, βαz+yα

### [ ]

x z, β, which we will extensively used.

During the past few decades, many authors have studied derivations in the context of prime and semiprime rings and Γ-rings with involution [11]-[14]. The notion of derivation and Jordan derivation on a Γ-ring were de-fined by [15]. Let M be Γ-ring. An additive mapping d M: →M is called a derivation if

### ( )

d x yα =d x αy+x d yα for all

and

### α

∈ Γ. An additive mapping d M: →M is called a

Jordan derivation if d x x

α

=d x

αx+x d xα

for all xM and

### α

∈ Γ.

Definition 1 [16]. An additive mapping

x xα

x xα

### )

* on a Γ-ring M is called an involution if

x yα

* =yx* and

**

### )

x xα = x xα for all

and

### α

∈ Γ. A Γ-ring M equipped with an involu-tion is called a Γ-ring M with involution.

Definition 2. An element x in a Γ-ring M with involution is said to be hermitian if *

x =x and skew-hermi-

tian if x*= −x. The sets of all hermitian and skew-hermitian elements of M will be denoted by H M and

### ( )

S M , respectively.

Example 1. Let F be a field, and D2

### ( )

F be a set of all diagonal matrices of order 2, with respect to the usual operation of addition and multiplication on matrices and the involution * on D2

### ( )

F be defined by

0 0 * : 0 0 a a a F a a     = ∈        

  with

0 0

* : ,

0 0

a a

a a F

a a

  − 

= − ∈

   

   

  , then we get

*

x =x and x*= −x. Definition 3. An additive mapping d M: →M is called a

### Γ

*-derivation if d x y

=d x

y*+x d y

for all

and

### α

∈ Γ.

To further clarify the idea of

### Γ

*-derivation, we give the following example.

Example 2. Let R be a commutative ring with characteristic of R equal 2. Define : , 0

a b

M a b R

a      =  ∈     

and 0 :

0 R α α α     

Γ =  ∈ 

 

 , then M and Γ are abelian groups under addition of matrices and M is a Γ-ring

under multiplication of matrices.

Define a mapping d M: →M by 0

0 0 0

a b b

d a     =           .

To show that d is a

### Γ

*-derivation, let

0 a b x a   =  

 , 0

c d y c   =    , * , 0 c d y c −   =     then

### )

0

0 0 0 0 0

0

.

0 0 0

a b c d a b c d

d x y d d

a c a c

a c a d b c a d b c

d

a c

α α α

α

α α

α α α α α

(3)

Now,

*

### ( )

0 0 0 0

0 0 0 0 0 0 0 0

0 0

0 0 0 0 0 0

0 0 0

,

0 0 0 0 0 0

b c d a b d

d x y x d y

c a

b c d a b d

c a

b c a d a d b c

α α

α α

α α

α α α

α

α α α α

           

+ =      +    

           

       

=    +  

       

− −

     

=  +  =

     

since char R

### ( )

=2, this implies that

*

### ( )

0

0 0

a d b c

d x αy +x d yα =  α + α 

  then

*

d x y

=d x

y +x d y

. Hence, d is a

### Γ

*-derivation.

Definition 4. An additive mapping d M: →M is called Jordan

*-derivation if

*

d x x

=d x

x +x d x

for all xM and

∈ Γ.

Every

### Γ

*-derivation is a Jordan

### Γ

*-derivation, but the converse in general is not true as shown by the fol-lowing example

Example 3. Let M be a Γ-ring with involution and let aM such that a aΓ =

0 and

for all

xM and

, but

for some

### M

such that xy.

Define a mapping d M: →M by d x

=x a

+a x

* for all xM and

### α

∈ Γ. To show that d is a Jordan

### Γ

*-derivation, we have on the one hand that

*

*

*

*

### )

* * * *

d x x d x x x d x x a a x x x x a a x

x a x a x x x x a x a x

α α α α α α α α α

α α α α α α α α

= + = + + +

= + + +

(3)

for all a x, ∈M and

### α

∈ Γ. On the other hand,

### )

* * *

d x xα = x xα αa+aα αx x =x x aα α +a xα αx (4)

for all a x, ∈M and

### α

∈ Γ. If we compare Equations ((3), (4)), we get

* * * * * *

x a xα α +a xα αx +x x aα α +x a xα α =x x aα α +a xα αx

then after reduction we get that d is a Jordan

### Γ

*-derivation. Now to show that d is not a

### Γ

*-derivation, we have on the one hand that

*

*

*

*

### )

* * * *

d x y d x y x d y x a a x y x y a a y

x a y a x y x y a x a y

= + = + + +

= + + + (5)

for all

and

### α

∈ Γ. On the other hand

### )

* * *

d x yα = x yα αa+aα αx y =x y aα α +a yα αx (6)

for all

and

### α

∈ Γ. If we compare Equations ((5), (6)), we get

* * * * * *

x a yα α +a xα αy +x y aα α +x a yα α =x y aα α +a yα αx

then after reduction we get that d is not a

### Γ

*-derivation.

In this paper we will prove that if a non-zero Jordan

### Γ

*-derivation d of a 2-torsion free semiprime Γ-ring M with involution satisfies d x

,xαZ M

for all xM and

∈ Γ, then d x

,xα =0.

*

α

### with Involution

To prove our main results we need the following lemmas.

(4)

hH M and

∈ Γ.

Proof. We have

,

d x x αZ M

 

  (7)

for all xM and

∈ Γ. Putting

### α

for x in (7), we get

,

d x xα x xα αZ M

 

  (8)

for all xM and

∈ Γ. Therefore,

*

### ( )

,

d x x x d x x x Z M

α

 +  ∈

 

for all xM and

### α

∈ Γ. Setting x= ∈h H M

### ( )

in the above relation, we get

,

### ( )

d h αh+h d h h hα α αZ M

 

  (9)

for all hH M

and

∈ Γ, because of

2

,

### ( )

d h αh+h d hα = h d hα − h d hα (10)

for all hH M

and

### α

∈ Γ. According to (9) and (10), we get

### ( )

2h d h h d h, α,h h Z M

α

 −  ∈

  (11)

for all hH M

and

### α

∈ Γ. Then from relation (11)

### ( )

2h d h ,h h α h d h, α,h h Z M

α

α α − α  ∈

   

    

### ( )

2h d h ,h h α h d h, α,h h Z M

α

α α + α  ∈

   

    

for all hH M

and

∈ Γ. Since d h

,hαZ M

, then d h h

, α,h h 0

α

### α

  =

  , and hence from

the above relation

### ( )

2h d h h hα , α α =2 hα d h h h, α α+ h h h, α ααd hZ M

   

   

for all hH M

and

∈ Γ. Therefore,

### ( )

2 , 2 , ,

2 , 2 ,

2 , 2 ,

h d h h h h h h d h h d h h

h h h d h h h d h h

h h h d h h h h d h Z M

α α α

α α

α α

α α α α α

α α α α

α α α α

= +

     

     

=  + 

=  + 

for all hH M

and

### α

∈ Γ. Then by using assumption (A), we obtain

### ( )

4h hα γh d h, αZ M

(12)

for all hH M

and

. And

### ( )

4 h h h d h, α,d h 0

β

   =

  (13)

for all hH M

and

### , ,

. Then from (13),

## )

### ( )

4 , , , ,

4 , ,

4 , , ,

4 , , 4 , ,

4 , , 4 , , 0

h h h d h d h h h d h h d h

h h d h h d h

h h d h h d h h h d h

h h d h h d h h d h h h d h

h h d h h d h h d h h h d h

α β α β

α β

α α β

α β α β

α β α β

α γ α γ

α γ

α α γ

α γ α γ

α γ α γ

(5)

for all hH M

and

. Since h d h,

αZ M

### ( )

, then from the above relation

### ( )

4 , , 4 , ,

4 , , 4 , , 0,

h h d h h d h h d h h d h h

h d h h d h h h h d h h d h

α β α β

α β α β

α γ α γ

α γ α γ

+

       

       

=  +     =

hence by using assumption (A), we obtain

### ( )

8hγh d h, αβh d h, α =0

(14)

for all hH M

and

. Therefore,

### ( )

8 h h d h, α h d h, α,d h 0

α

   =

  (15)

for all hH M

and

### , ,

. Then from relation (15),

### )

8h h d h, α h d h, α ,d h 8 h d h, α h d h, α h d h, α 0

α

 

 + 

=

for all hH M

and

### , ,

. By using relation (15) and assumption (A), we get

### ( )

8 h d h, αβh d h, α δh d h, α =0 (16) for all hH M

and

### , ,

. Since M is 2-torsion free, we get

### (

h d h, αβh d h, α

δh d h,

### ( )

α =0 (17) for all hH M

and

### , ,

. Right multiplication of (17) by zδh d h,

### ( )

α and using assumption (A), we get

### (

h d h, αβh d h, α

δ δz h d h,

αβh d h,

### ( )

α =0 (18) for all hH M

and

### , ,

. By semiprimeness of M, we have

### ( )

, , 0

h d h αβ h d h α =

   

    (19)

for all hH M

and

### , ,

. Left multiplication of (19) by z yields

,

### ( )

, 0

d h h αβ βz d h h α =

   

    (20)

for all hH M

and

### , ,

. By semiprimeness of M again, we get d h h

### ( )

, α =0 for all hH M

and

### , ,

. 

Lemma 2 Let M be a 2-torsion free semiprime Γ-ring with involution and d M: →M be a Jordan

### Γ

*- derivation which satisfies d x

,xαZ M

for all xM and

∈ Γ, then d s

,sα =0 for all

sS M and

### α

∈ Γ.

Proof. Putting x+y for x in (7),

,

,

,

,

,

### ( )

d x+y x+y α = d x x α+ d x y α + d y x α+ d y y αZ M

         

         

for all

and

### α

∈ Γ. By using Lemma (1), we get

d x ,yα+d y ,xα

Z M

(21) for all

and

### α

∈ Γ. Replacing x by

and y by *

x yields

* *

### ( )

, ,

d x x x α d x x x Z M

α

α  α 

  + ∈

    (22)

for all xM and

### α

∈ Γ. Setting x= ∈s S M

, we get

* *

### ( )

, ,

d s s s d s s s Z M

α α

α  α 

  + ∈

    (23)

(6)

*

,

### ( )

d s sα =d s αs +s d sα =s d sα −d s αs=s d sαZ M

(24)

for all sS M

and

### α

∈ Γ. So then from (23) and (24), we get

* *

### ( )

, , ,

s d s α s d s s s Z M

α  α α

  + ∈

   

for all sS M

and

∈ Γ. Hence,

### (

d s ,s α,s s s d s, α

Z M

α

  +

 

for all sS M

and

∈ Γ. Since d s

,sαZ M

, then d s s

### ( )

, α,s 0

α

  =

  , hence from the above

relation,

,

s s d sα αZ M

 

 

(25)

for all sS M

and

∈ Γ. Therefore,

### ( )

, , , 2 ,

s s d sα α =sα s d s α + s d s ααs= sα s d s αZ M

       

        (26)

for all sS M

and

### α

∈ Γ. Since s d s,

αZ M

, we obtain

### ( )

2 s s d s, α,d s 0

α

   =

  (27)

for all sS M

and

### α

∈ Γ. Then from relation (27),

### )

2 s s d s, α,d s s d s, α s d s, α 0

α

 +

  =

for all sS M

and

### α

∈ Γ. By using relation (27) again,

### ( )

2s d s, αγs d s, α =0 (28)

for all sS M

and

### α

∈ Γ. Since M is 2-torsion free, we get

,

### ( )

, 0

d s s αγ d s s α =

   

   

(29)

for all sS M

and

### α

∈ Γ. Right multiplication by z yields

,

### ( )

, 0

d s s αγ γz d s s α =

   

   

(30)

for all sS M

and

### α

∈ Γ. By semiprimeness of M, we therefore get d s

### ( )

,sα =0 for all sS M

and

### α

∈ Γ. 

Remark 1 [17]. A Γ-ring M is called a simple Γ-ring if M MΓ ≠0 and its ideals are 0 and M.

Remark 2. Let M be a 2-torsion free simple Γ-ring with involution, then every xM can be uniquely represented in the form 2x= +h s where hH M

and sS M

### ( )

.

Proof. Define H M

=

xM x; * =x

, S M

=

xM x; * = −x

### }

, since 2M is an ideal of M and M is

sim-ple, it implies that 2M=M. So for every xM, x 2 makes sense and so we can write

* *

2 2

x x x x

x= + + −

Now

### )

*

* *

* *

* * * * *

1 1 1 1

2 2 2 2 2 2

x x x x

x x x x x x x x

 +  = + = + = + = + = +

 

 

hence

*

2

x x

H M

+

(7)

### )

*

* * *

*

* * * * *

1 1 1 1

2 2 2 2 2 2

x x x x

x x x x x x x x

 −  = = = = − = −

 

 

hence

*

2

x x

S M

Therefore

### ( )

* *

2 2

x x x x

x= + + − ∈H M +S M

hence M =H M

+S M

. Let xH M

S M

, then xH M

and xS M

### ( )

, so x*=x and

*

x = −x. Therefore

### x

which implies that 2x=0, so x=0. Thus H M

S M

=0. Hence

### ( )

2M=M=H M +S M implies that 2x= +h s where hH M

and sS M

### ( )

.  Theorem 1. Let M be a 2-torsion free semiprime Γ-ring with involution and d M: →M be a Jordan

### Γ

*- derivation which satisfies d x

,xαZ M

for all xM and

∈ Γ and d h

,sαZ M

or

,

d s h αZ M

 

  for all hH M

, sS M

and

### α

∈ Γ, then [d(x),x]α=0 for all xM and

### α

∈ Γ.

Proof. Assume that d h

,sαZ M

for all hH M

, sS M

and

### α

∈ Γ. By using Lemma (1), we have

### ( )

, 0

d h h α =

 

  (31) for all hH M

and

### α

∈ Γ. For h h1, 2H M

### ( )

, putting h1+h2 for h in (31) yields

1 2

, 1 2

1 , 1

1 , 2

2 , 1

### ( )

2 , 2 0

d h +h h +h α = d h h α+ d h h α + d h h α+ d h h α =

         

         

for all hH M

and

### α

∈ Γ. By using relation (31), we obtain

1 , 2

### ( )

2 , 1 0

d h h α+ d h h α =

   

    (32)

for all h h1, 2H M

and

### α

∈ Γ. Since s sβ ∈H M

for all sS M

### ( )

, then replace h2 by

in (32), to get

1 ,

### )

, 1 0

d h s sβ α+ d s sβ h α =

   

    (33)

for all h1H M

, sS M

and

### α

∈ Γ. By using Lemma (2), we have

*

### ( )

, 0

d s s d s s s d s s d s d s s s d s

β

=

+

=

= = (34)

for all sS M

and

### β ∈Γ

. According to relations (33) and (34), we get

1 , ,

, 1

1 ,

### ( )

, , 1 0

d h s s α s d s β h d h s s α d s s β h

α α

β +  = β   =

       

       

for all sS M

and

### β ∈Γ

. By using Lemma (2), we get

1 , 0

d h s sβ α =

 

  (35)

for all sS M

and

### β ∈Γ

. Then from relation (35),

1 ,

### ( )

1 , 0

sβd h sα+d h sαβs=

for all h1H M

, sS M

and

### α

∈ Γ. Therefore since d h

1 ,sαZ M

, we obtain

### ( )

1

2sβd h ,sα =0 (36)

for all h1H M

, sS M

and

∈ Γ. Hence,

1

### ( )

1

2 d h ,s d h ,s 0

α γ

   =

(8)

for all h1H M

, sS M

and

. Therefore,

1

1

1

1

1

1

## )

2 d h ,s d h ,s α 2 s d h , d h ,s α d h ,s β d h ,s β 0

γ γ

β β γ

   =    +  =

   

for all h1H M

, sS M

and

### , ,

. Then by using (37), we get

1

### ( )

1

2d h ,sβγd h ,sβ =0 (38)

for all h1H M

, sS M

and

### ,

. Since M is 2-torsion free, we get

1 ,

### ( )

1 , 0

d h s βγ d h s β =

   

   

(39)

for all h1H M

, sS M

and

### ,

. Right multiplication of relation (39) by z yields

1 ,

### ( )

1 , 0

d h s βγ γz d h s β =

   

    (40)

for all h1H M

, sS M

and

### ,

. By semiprimeness of M, we get

1 , 0

d h s α =

 

 

(41)

for all h1H M

, sS M

and

### α

∈ Γ. Putting s for x and h for y in relation (21), we get

,

,

### ( )

d s h α+ d h s αZ M

   

    (42)

for all hH M

, sS M

and

### α

∈ Γ. Comparing relations (41) and (42), we get

,

d s h αZ M

 

 

(43)

for all hH M

, sS M

and

### α

∈ Γ. Since h hγ ∈H M

for all hH M

### ( )

, then from relation (43), we obtain

,

d s h hγ αZ M

 

 

(44)

for all hH M

, sS M

and

∈ Γ. Then

,

,

,

### ( )

d s h hγ α =hγ d s h α+ d s h αγhZ M

     

      (45)

for all hH M

, sS M

and

∈ Γ. Since d s

,hαZ M

for all hH M

, sS M

and

### α

∈ Γ, then from relation (45), we get

### ( )

2hγd s ,hαZ M (46)

for all hH M

, sS M

and

. Hence

### ( )

2 d s h, d s h, 0

α β

   =

  (47)

for all hH M

, sS M

and

### , ,

. Then from relation (47),

## )

2 h d s , d s ,h α d s ,h α d s ,h β 0

β

γ  + γ  =

for all hH M

, sS M

and

### , ,

. Then by using (47), we get

### ( )

2d s ,hβγd s ,hβ =0 (48)

for all hH M

, sS M

and

### ,

. Since M is 2-torsion free, we get

,

### ( )

, 0

d s h βγ d s h β =

   

    (49)

for all hH M

, sS M

and

### ,

. Right multiplication of the relation (49) by z we get

,

### ( )

, 0

d s h βγ γz d s h β =

   

    (50)

(9)

, 0

d s h β =

 

 

(51)

for all hH M

, sS M

and

. To prove d x

### ( )

,xα =0, since M is 2-torsion free, we only show

### ( )

4d x ,xα =0 (52)

for all xM and

### α

∈ Γ. By using Remark 2, we have for all xM and

### α

∈ Γ that 2x= +s h for all

sS M and hH M

. Therefore

### )

4d x ,xα =d 2x , 2xα =d s+h ,s+hα =0

for all hH M

, sS M

and

∈ Γ. Hence

### ( )

4d x ,xα =d s ,sα +d h ,sα+d s ,hα+d h h, α =0 (53)

for all hH M

, sS M

and

### α

∈ Γ. By using Lemma (1), Lemma (2) and relation (41), (51), we get

, 0.

d x x α =

 

 

Now assume d s

,hαZ M

for all hH M

, sS M

and

### α

∈ Γ. Since h hγ ∈H M

for all

### ( )

hH M , then we get

,

d s h hγ αZ M

 

  (54)

for all hH M

, sS M

and

### ,

. Then from relation (54), we obtain

,

,

,

### ( )

d s h hγ α =hγ d s h α+ d s h αγhZ M

     

      (55)

for all hH M

, sS M

and

∈ Γ. Since d s

,hαZ M

for all hH M

, sS M

and

### α

∈ Γ, then from relation (55), we get

### ( )

2hγd s ,hαZ M (56)

for all hH M

, sS M

and

. Hence

### ( )

2 d s h, d s h, 0

α β

   =

  (57)

for all hH M

, sS M

and

### , ,

. Then from relation (57),

## )

2 h d s , d s ,h α d s ,h α d s ,h β 0

β

γ  + γ  =

for all hH M

, sS M

and

### , ,

. Then by using relation (57) we get

### ( )

2d s ,hβγd s ,hβ =0 (58)

for all hH M

, sS M

and

### ,

. Since M is 2-torsion free, we get

,

### ( )

, 0

d s h βγ d s h β =

   

    (59)

for all hH M

, sS M

and

### ,

. Right multiplication of relation (59) by z yields

,

### ( )

, 0

d s h βγ γz d s h β =

   

    (60)

for all hH M

, sS M

and

### ,

. By semiprimeness of M, we get

, 0

d s h α =

 

 

(61)

for all hH M

, sS M

and

### α

∈ Γ. Comparing relations (42) and (61), we get

,

d h s αZ M

 

  (62)

(10)

,

d h s sγ αZ M

 

  (63)

for all hH M

, sS M

and

∈ Γ. Then,

,

,

,

### ( )

d h s sγ α =sγ d h s α+ d h s αγsZ M

     

     

(64)

for all hH M

, sS M

and

∈ Γ. Since d h

,sαZ M

for all hH M

, sS M

and

### α

∈ Γ, then from relation (64), we get

### ( )

2sγ d h ,sαZ M (65)

for all hH M

, sS M

and

. Hence

### ( )

2 d h s, d h s, 0

α β

   =

  (66)

for all hH M

, sS M

and

### , ,

. Then from relation (66)

## )

2 s d h , d h ,s α d h ,s α d h ,s β 0

β

γ  + γ  =

for all hH M

, sS M

and

### , ,

. Then by using (66) we get

### ( )

2d h ,sβγd h ,sβ =0

(67)

for all hH M

, sS M

and

### ,

. Since M is 2-torsion free, we get

,

### ( )

, 0

d h s βγ d h s β =

   

    (68)

for all hH M

, sS M

and

### ,

. Right multiplication of relation (68) by z yields

,

### ( )

, 0

d h s βγ γz d h s β =

   

    (69)

for all hH M

, sS M

and

### ,

. By semiprimeness of M, we get

, 0

d h s β =

 

 

(70)

for all hH M

, sS M

and

### β ∈Γ

. Therefore, by using Lemma (1), Lemma (2) and relation (61), (70), we get a similar result as the first assumption d x

### ( )

,xα =0 for all xM and

### α

∈ Γ, and hence the proof of the theorem is complete. 

### Acknowledgements

This work is supported by the School of Mathematical Sciences, Universiti Sains Malaysia under the Short-term Grant 304/PMATHS/6313171.

### References

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http://dx.doi.org/10.1307/mmj/1029000167

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