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Left Generalized Derivations on Prime gamma Rings

January 2018
Annals of Pure and Applied Mathematics 16(1)

Authors:

Sri Venkateswara University

Let be a prime-ring with 2-torsion free, a nonzero ideal of M and : → a left generalized derivation of , with associated nonzero derivation d on. If () ∈ () for all ∈ , then is a commutative-ring.

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Annals of Pure and Applied Mathematics

Vol. 16, No. 1, 2018, 127-131

ISSN: 2279-087X (P), 2279-0888(online)

Published on 5 January 2018

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/apam.v16n1a14

127

Annals of

Left Generalized Derivations on Prime Γ-Rings

C. Jaya Subba Reddy

, K. Nagesh

and A. Sivakameshwara Kumar

Department of Mathematics, Sri Venkateswara University

Tirupati-517502, India. E-mail: cjsreddysvu@gmail.com

Department of Mathematics, Rayalaseema University

Kurnool, Andhrapradesh, India. E-mail: nagesh.kunda100@gmail.com

Department of Mathematics, Rayalaseema University

Kurnool, Andhrapradesh, India. E-mail: kamesh1069@gmail.com

Received 6 December 2017; accepted 26 December 2017

Abstract. Let  be a prime -ring with 2-torsion free,  a nonzero ideal of M and

    a left generalized derivation of , with associated nonzero derivation d on .

If    for all   , then  is a commutative -ring.

Keywords: Gamma ring, prime gamma ring, derivation, generalized derivation, left

generalized derivation, commutators.

AMS Mathematics Subject Classification (2010): 06F25

1. Introduction

The notion of -ring was first introduction by Nobusawa [9] and also shown that -ring,

more general than rings. Barnes [1] slightly weakened the conditions in the definitions of

a -rings in the sense of Nobusawa. After the study of -rings by Nobusawa [9] and

Barnes [1], many researchers have a done lot of work and have obtained some

generalizations of the corresponding results in ring theory [6][8]. Barnes [1] and kyuno

[8] studied the structure of -ring and obtained various generalizations of the

corresponding results of ring theory. Hvala [4] introduced the concept of Generalized

derivations in rings. Dey, Paul and Rakhimov [3] discussed some properties of

Generalized derivations in semiprime gamma rings Bresar [2] studied on the distance of

the composition of two derivations to the generalized derivations. Jaya Subba Reddy. et

al. [5] studied centralizing and commutating left generalized derivation on prime ring is

commutative. Jaya Subba Reddy et al. [12] studied some results of symmetric reverse

bi-derivations on prime rings, Ozturk et al. [10] studied on derivations of prime gamma

rings. Khan et al. [6,7] studied on derivations and generalized derivations on prime -

rings is a commutative. In this paper we extended some results on left generalized

derivations on prime -ring is a commutative.

2. Preliminaries

If  and are additive abelian groups and there exists a mapping       

which satisfies the following conditions:

For all     and    ,

C. Jaya Subba Reddy, K. Nagesh and A. Sivakameshwara Kumar

128

(i)   , denoted by , is an element of 

(ii)       ,       ,      

(iii)   

then  is called a -ring [1]. It is known that from (i), (iii) the following follows:

       (A)

for all  and  in  and all  in [1].

Every ring is a -ring with   . However a -ring need not be a ring. Let  be a -

ring, then  is called a prime -ring, if    implies    or   , for all

    and  is called a semiprime -ring, if    implies   , for all  

. Every prime-ring is obviously semiprime. If  is a -ring, then  is said to be 2-

torsion free if    implies    , for all    . An additive subgroup  of  is

called a left (right) ideal of  if      . If  is both left and right ideal of ,

then we say  is an ideal of . Moreover, the set       

  s called the centre of the -ring . We shall write  



   , for

all     and    . We shall make use of the basic commutator identities:

 



  



   



and  



  



   



, for all

     and    . If -ring satisfies the assumption (B)   , for all

     and    . Let  be a -ring. An additive mapping     is called a

derivation on  if     holds for all      and   . An

additive mapping     is called a generalized derivation if there exists a derivation

    such that     holds for all     and   . An

additive mapping     is called a left generalized derivation if there exists a

derivation      such that    holds for all     and

  . A derivation of the form      where   are fixed elements of  and

   is called generalized inner derivation. An additive mapping     is called a

left (right) centralizer if     for all     and

  .

Lemma 2.1. Let  be a prime -ring with 2-torsion free and  a nonzero ideal of . Let

    be a left generalized derivation of , associated with derivation d. If 

, for all   , then   .

Proof: For all     and   ,  . That is,    , which

implies    . Let       . The last relation along with (A) gives,

  . Since  is prime -ring and  is a nonzero ideal, so  , for all

  . Hence , by hypothesis,  , for all   , and   , and   . That is,

   , which gives   Let       . The last relation

along with (A), implies  . Since  is nonzero and primeness of , gives

  .

Lemma 2.2. Let  be a nonzero ideal of a prime -ring ,    and    is a left

generalized derivation of , with associated nonzero derivation , then

(i) If    for all    and   , then   ,

(ii)If   for all    and   , then   .

Proof:

Left Generalized Derivations on Prime Γ-Rings

129

(i) For any       and   ,  . That is,

     Which implies,   . Since  is a

nonzero ideal of  and   , we get   .

(ii) Proof is similar to (i).

3. Main results

Theorem 3.1. Let  be a prime -ring with 2-torsion free and  a nonzero ideal of .

Let     be a left generalized derivation of , with associated nonzero derivation d

on . If    for all   , then  is a commutative -ring.

Proof: Using hypothesis, we have  



 , for all    ,   , which

gives   



 

 



 



  



  



  

Using hypothesis, we get

 



  



  

        

Using (B), from the last equation we get

        

    , for all     ,    

. (1)

Let   . Replacing  by  in equation (1), we get

    

Which along with equation (1) and (B) gives ,

    

 



 , for all     and    .

Since  is a nonzero ideal of  and   , therefore  is a commutative - ring.

Theorem 3.2. Let  be a prime -ring with 2-torsion free and  a nonzero ideal of .

Let     be a generalized derivation and left generalized derivation of , with

associated derivation d on . If    and  



 , for all   ,    , then

either    or  .

Proof: Using hypothesis, we have

 



  for any   ,    and    .

This gives   



 

 



  



 

The last equation gives

 



 



  



  



  

Using hypothesis, from the last equation we get

 



 



  



 

         

Using (B), from the last equation we get

         

        (2)

Let   . Replacing  by  in equation (2), we get

       

        

 

Using equation (2), from the last equation we get

C. Jaya Subba Reddy, K. Nagesh and A. Sivakameshwara Kumar

130

       



 

       



 

Using (B), from the last equation we get

   



 

 



   



 

Replacing  by  from the last equation, we get

 



   



 

 



  , for all   ,    and     .

Since  is nonzero ideal of prime -ring , therefore either   or   .

Corollary 3.2.1. Let  be a prime -ring with 2-torsion free and  a nonzero ideal of

. Let     be a left generalized derivation of , with associated derivation d on

. If  



 , for all    ,   , then  is a commutative -ring.

Proof: Using Theorem 3.2, we have   , we get the corollary 3.2.1 proof.

Theorem 3.3. Let  be a prime -ring with 2-torsion free and  a nonzero ideal of .

Let     be a left generalized derivation of , with associated derivation d on . If

  , for all    ,   , then   .

Proof:   , for any    ,   .

   (3)

Let   ,   .Then replacing  by  in equation (3), we get

  

  , for all    , and    .

Since  is a nonzero ideal of the prime -ring , therefore either     for all

   or   for all   . If    , then   for all   . Replacing

 by  in the last equation, we get  , which implies 

  , which gives     . That is   , for all     ,

  . Thus   for all    for both cases. So   .

Theorem 3.4. Let  be a prime -ring with 2-torsion free and  a nonzero ideal of .

Let     be a left generalized derivation of , with associated derivation d on . If

  , for all    ,   , then   .

Proof:    , for all    ,   .

    (4)

Let   ,   . Replacing  by  in equation (4), we get

   

    

Using (B), from the last equation we get

    

  (5)

Let   ,   . Then replacing  by , we get

 

Using equation (5) in above equation, we get

 

Using (B), from the last equation we get

Left Generalized Derivations on Prime Γ-Rings

131

   

 



 

Since  is a nonzero ideal of the prime -ring . Therefore either   for all   

or  



  for all     and    . Let         and  

    



      . Obviously  and  are additive subgroups of .

Moreover  is the set theoretic union of  and . But a group cannot be set theoretic

union of two proper sub groups . Hence either    or    . If    , we have

 , which completes the proof. If   , then    



  



for all     ,    and    . Thus, we obtain   , Using Theorem

3.2, we get   .

Acknowledgements. We are thankful to the reviewers for their comments to improve the

presentation of the paper.

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On the Γ-rings of Nobusawa

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N. Nobusawa recently introduced the notion of a Γ-uring, more general than a ring, and obtained analogues of the Wedderburn theorems for Γ-rings with minimum condition on left ideals. In this paper the notions of Γ-homomorphism, prime and (right) primary ideals, m-systems, and the radical of an ideal are extended to Tarings, where the defining conditions for a Γ-ring have been slightly weakened to permit defining residue class Γ-rings. The radical R of a Γ-ring M is shown to be an ideal of M, and the radical of M/R to be zero, by methods similar to those of McCoy. If M satisfies the maximum condition for ideals, the radical of a primary ideal is shown to be prime, and the ideal Q ≠ M is P-primary if and only if Pn ⫅ Q for some n, and AB ⫅ Q, A ⫋ P implies B ⫅ Q. Finally, in Γ-rings with maximum condition, if an ideal has a primary representation, then the usual uniqueness theorems are shown to hold by methods similar to those of Murdoch.

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GLASGOW MATH J

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A well-known theorem of E. Posner [10] states that if the composition d 1 d 2 of derivations d 1 d 2 of a prime ring A of characteristic not 2 is a derivation, then either d 1 = 0 or d 2 = 0. A number of authors have generalized this theorem in several ways (see e.g. [1], [2], and [5], where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in [3] (resp. [12]). Recently, M. Mathieu [8] extended Posner's theorem to arbitrary C * -algebras.

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