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International Journal of Mathematical Archive-8(7), 2017, 91-94
Available online through www.ijma.info ISSN 2229 – 5046
International Journal of Mathematical Archive- 8(7), July – 2017 91
PRIME GAMMA RINGS
WITH CENTRALIZING AND COMMUTING LEFT GENERALIZED DERIVATIONS
A. SIVAKAMESHWARA KUMAR1, C. JAYA SUBBA REDDY*2
1Research Scholar, Rayalaseema University, Kurnool, Andhra Pradesh, India.
2Department of Mathematics, S. V. University, Tirupati, Andhra Pradesh, India.
(Received On: 20-06-17; Revised & Accepted On: 22-07-17)
ABSTRACT
Let be a prime Γ-ring satisfying a certain assumption and a nonzero derivation on M. Let : be a left
generalized derivation such that is centralizing and commuting on a left ideal of . Then we prove that is
commutative.
Key words: Prime Γ-ring, Centralizing and Commuting, Derivation, Left derivation, Generalized derivations, Left
generalized derivations.
PRELIMINARIES
Let and Γ be additive abelian groups. If there exists a mapping (,,) of ×× , which satisfies
the conditions
(i)
(ii) (+) = +, (+)= +, (+)= +
(iii) () =() for all ,, and , , then is called a Γ-ring.
Every ring is a Γ-ring with =Γ. However a Γ-ring need not be a ring. Let be a Γ-ring. Then an additive
subgroup of is called a left (right) ideal of if ( ). If is both a left and a right ideal, then
we say is an ideal of . Suppose again that is a Γ-ring. Then is said to be a 2-torsion free if 2= 0 implies
= 0 for all . An ideal
of a Γ-ring is said to be prime if for any ideals and of ,
implies
or
. An ideal of a Γ-ring is said to be semiprime if for any ideal of , implies
. A Γ-ring is said to be prime if = (0) with , , implies = 0 or = 0 and semiprime if
= (0) with implies = 0 . Furthermore, is said to be commutative Γ-ring if = for all
, and . Moreover, the set ()= { : = for all and } is called the centre of the
Γ-ring . If is a Γ-ring, then [,]= is known as the commutator of and with respect to , where
, and . We make the basic commutator identities:
[,]= [,] +[,] and [,]= [,] +[,] , for all , and . We consider
the following assumption:
().............. =, for all ,, and , . An additive mapping : is called a derivation if
()=() +() holds for all , and . A mapping is said to be commuting on a left ideal
of if [(),]= 0 for all and and is said to be centralizing if [(),] ()for all and
. An additive mapping : is said to be a generalized derivation on , if ()=() +()
holds for all , and , where is a derivation on . An additive mapping : is called a left
generalized derivation on , if ()=()+() holds for all , and , where is a derivation
on .
Corresponding Author: C. Jaya Subba Reddy*2
2Department of Mathematics, S. V. University, Tirupati, Andhra Pradesh, India.
A. Sivakameshwara Kumar1, C. Jaya Subba Reddy*2 /
Prime Gamma Rings with Centralizing and Commuting Left Generalized Derivations / IJMA- 8(7), July-2017.
© 2017, IJMA. All Rights Reserved 92
INTRODUCTION
The concept of the -ring was first introduced by Nobusawa[13] and also shown that -rings, more general than rings.
Bernes [1] weakened slightly the conditions in the definition of -ring in the sense of Nobusawa. Bresar[2] studied
centralizing mappings and derivations in prime rings. Kyuno[9], Luh[10], Hoque and Paul[5], [6] and others were
obtained a large numbers of important basic properties of -rings in various ways and determined some more
remarkable results of -rings. Ceven[3] studied on Jordan left derivations on completely prime -rings. Mayne[12]
have developed some remarkable result on prime rings with commuting and centralizing. Jaya subba reddy.C et.al [8]
studied centralizing and commutating left generalized derivation on prime ring is commutative. Hoque and paul [7]
studied prime gamma rings with centralizing and commuting generalized derivations is a commutative. In this paper,
we extended some results on prime gamma rings with centralizing and commuting left generalized derivations is a
commutative.
Some preliminary results
We have to make some use of the following well-known results
Remark 1: Let be a prime -ring. If () with 0 (), then ().
Remark 2: Let be a prime -ring and a nonzero left ideal of . If is a nonzero derivation on , then is also a
nonzero on .
Remark 3: Let be a prime -ring and a nonzero left ideal of . If is commutative, then is also commutative.
Lemma 1: Suppose is a prime -ring satisfying the assumption () and : be a derivation. For an element
, if ()= 0, for all and , then either = 0 or = 0.
Proof: By our assumption, ()= 0, for all , and .
Replacing by in above equation, we get
()= 0
(() +()) = 0
() +()= 0
()= 0, for all , , and , .
If is not a zero, that is, if ()0, for some .
By definition of prime -ring, then = 0. Hence proved.
Lemma 2: Suppose is a prime -ring satisfying the assumption () and a nonzero left ideal of . If has a
derivation which is zero on , then is zero on .
Proof: By the hypothesis, ()= 0
Replacing by J in above equation then, we get
( J)= 0
()J + MD(J)= 0
()J = 0.
By Lemma 1, must be zero, since is nonzero.
Lemma 3[7]: Suppose is a prime -ring satisfying the assumption () and a nonzero left ideal of . If is
commutative on , then is commutative.
Lemma 4: Suppose is a prime -ring and : be a additive mapping. If is centralizing on a left ideal of
, then () (), for all ().
Proof: is a centralizing a on left ideal of , we have [(),] () for all and .
By linearization, we have
, + , for all .
[(+),+] ()
A. Sivakameshwara Kumar1, C. Jaya Subba Reddy*2 /
Prime Gamma Rings with Centralizing and Commuting Left Generalized Derivations / IJMA- 8(7), July-2017.
© 2017, IJMA. All Rights Reserved 93
is a additive mapping then
[()+(),+] ()
[(),]+ [(),]+ [(),]+ [(),] ()
is a centralizing on left ideal of then, we get
[(),]= 0, [(),]= 0
[(),]+ [(),] (). (1)
If (), then equation (1) becomes
[(),] ().
Replacing by () in above equation then, we get
[(),()] ()
[(),()] + ()[(),] ()
()[(),] (). If [(),]= 0.
Then () ().
The centralizer of in and hence () (). Otherwise, if [(),]0, remark 1 follows that () ().
Hence the lemma.
Theorem 1: Let be a prime -ring satisfying the assumption () and a nonzero derivation on . If is a left
generalized derivation on a left ideal of such that is commuting on , then is commutative.
Proof: Since is commuting on , we have
[(),]= 0, for all and .
Replacing by + in above equation, we get
[(+),+]= 0
[()+(), +]= 0
[(),]+ [(),]+ [(),]+ [(),]= 0
[(),]+ [(),]= 0 (2)
Replacing by in equation (2), we get
[(),]+ [(),]= 0
[(),] + [(),]+ [()+(),]= 0
[(),] + [(),]+ [(),]+ [(),]= 0
[(),] + [(),]+[,]()+ a[(),]+ [(),]= 0
is centralizer then, [(),] = 0, [,]() = 0.
[(),]+ a[(),]+ [(),]= 0
( [(),]+ [(),]) + [(),]= 0
Using equation (2) in above equation, we get
[(),]= 0. (3)
Replacing by in above equation (3), we get
[(),]= 0
[(),] +()[,]= 0
[(),] +() [,] +() [,]= 0
() [,]= 0, for all , and ,,, .
Since is prime -ring, thus ()= 0 or [,]= 0.
Since is nonzero derivation on , then by lemma 2, is nonzero on .
Suppose ()0 for some , then ().
Let with (). Then ()= 0 and + (), that is, (+)= 0 and so ()= 0, which is a
contradiction. Thus () for all . Hence is commutative and hence by lemma 3, is commutative. Hence
the theorem.
A. Sivakameshwara Kumar1, C. Jaya Subba Reddy*2 /
Prime Gamma Rings with Centralizing and Commuting Left Generalized Derivations / IJMA- 8(7), July-2017.
© 2017, IJMA. All Rights Reserved 94
Theorem 2: Let be a prime -ring satisfying the assumption () and a left ideal of with ()0. If is
a left generalized derivation on with associated nonzero derivation such that is commuting on , then is
commutative.
Proof: we claim that, ()0 because of is commuting on and the proof is complete.
Now from equation (1), we get
[(),]+ [(),] ()
We replace by with 0 (), then we get
[(),]+ [(),] ()
[()+(),]+ [(),] + b[(),] ()
[(),]+ [(),]+ [(),] ()
[,]()+ [(),]+ [(),] +() [,]+ [(),] + [(),] ()
() [,]= 0 for all , [,]= 0
Since () is a centralizer on .
() () [(),]= 0.
[(),]+ [(),] +[(),] ()
From lemma 1, () () and hence [(),] +[(),] (). Since is a centralizing on , we have
[(),] () and consequently [(),] (). As is nonzero, remark 1 follows that
[(),] (). This implies is centralizing on and hence we conclude that is commutative.
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Source of support: Nil, Conflict of interest: None Declared.
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