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Right Ideals and Left Generalized Derivations on Prime Rings

December 2016

Authors:

C.Jaya Subba Reddy

Sri Venkateswara University

Dhananjaya Reddy

Government Degree College

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Open Journal of Applied & Theoretical Mathematics (OJATM)

Vol. 2, No. 4, December 2016, pp. 134~139

ISSN: 2455-7102

Journal homepage:

http://ojal.us/ojatm/

134

RIGHT IDEALS AND LEFT GENERALIZED DERIVATIONS ON

PRIME RINGS

C.JAYA SUBBA REDDY

A.SIVAKAMESHWARA KUMAR

DR.DHANANJAYA REDDY

Department of Mathematics, S.V.University,

Tirupati, AndhraPradesh,India.

Lecturer, Department of Mathematics, Govt.Degree College,

Puttur, Andhra Pradesh,India.

cjsreddysvu@gmail.com

kamesh1069@yahoo.com

djreddy65@gmail.com

Article Info

ABSTRACT

Let be a prime ring and be a left derivation on . If

f is a left generalized derivation on such that f is

commuting and centralizing on a right ideal I of ,

then is commutative.

Keyword:

Prime rings, Right ideals,

Left derivation,

Generalized derivation,

Generalized left

derivation, Centralizing

and commuting..

Open Journal of Applied & Theoretical Mathematics

(OJATM)

Preliminaries: Throughout this paper will denote an associative ring with centre

A ring is said to be a prime if implies that either or . An

additive mapping is said to be derivation if , for all

and an additive mapping is said to be left derivation if

A mapping is said to be commutating on a

left ideal of if , for all and is said to be centralizing if

,for all . An additive mapping said to be generalized

derivation if there exists a derivation such that , for all

. An additive mapping is said to be a left generalized derivation if

, for all , where is a left derivation on . The

commutator and anti commutator for all

Open Journal of Applied & Theoretical Mathematics (OJATM)

Vol. 2, No. 4, December 2016, pp. 134~139

ISSN: 2455-7102

Journal homepage:

http://ojal.us/ojatm/

135

and commutator identities , and for

all .

Introduction: The study of the commutativity of prime rings with derivations was

initiated by E.C.Posner[10]. Recently, M.Bresar [3, 4] define generalized derivation of

rings. Hvala [6] studied generalized derivation in prime rings. Golbasi [5] extended

some well known results concerning derivations of prime rings to the generalized

derivations and a nonzero left ideal of a prime ring which is semi prime as a ring. Jaya

Subba Reddy et.al [7, 8] studied centralizing and commutating left generalized

derivation on prime ring is commutative. Afrah Mohammad Ibraheem[1] studied right

ideal and generalized reverse derivation on prime rings is commutative. In this paper

we extended some results right ideals and left generalized derivations on prime ring is

commutative.

In order to prove the main results, first we prove the following lemma:

Remark (1) [2]: Let be a prime ring. For a nonzero element , if

, then .

Lemma (1): Let be a prime ring and be a left derivation on . For an element

if , for all , then either or .

Proof: For , let , for all .

(1)

Replace by in (1), we have , then

, for all

(2)

By using (1), we get , for all .

if , for some , then by definition of prime ring. Hence proved.

Lemma (2): Let I be a nonzero right ideal of a prime ring . If has a zero left

derivation

on I, then is also zero left derivation on .

Proof: Let is a right ideal or .

We assume that

(3)

Since we have:

(4)

By using (3), we get .

Since , then by lemma (1), .

Open Journal of Applied & Theoretical Mathematics (OJATM)

Vol. 2, No. 4, December 2016, pp. 134~139

ISSN: 2455-7102

Journal homepage:

http://ojal.us/ojatm/

136

Lemma (3) [9]: Let be a prime ring and a non zero right ideal of . If is

commutave, then is also commutative.

Theorem (1): Let be a prime ring and be a non zero right ideal of . If is a non

zero left generalized derivation on , such that is a centralizing on , then is

commutative.

Proof: Let be a centralizing on

, for all

(5)

Replacing by in (5), we get

(6)

Thus, , for all

and , for all .

(7)

Also and , for all .

(8)

We Replace by in (8) ,we get

,for all

(9)

Replacing by in (9), we get

, for all and .

Since is a prime, we get either or

If for all then by Lemma (1), this is a contradiction.

Open Journal of Applied & Theoretical Mathematics (OJATM)

Vol. 2, No. 4, December 2016, pp. 134~139

ISSN: 2455-7102

Journal homepage:

http://ojal.us/ojatm/

137

So , for all , that’s mean is commutative and hence by Lemma (3),

is commutative.

Theorem (2): Let be a prime ring and be a right ideal of . If f is left generalized

derivation on with a left derivation on , such that f is centralizing on , then for

all , .

Proof: Since f is centralizing on , we have

, for all . (10)

By linearizing (10), for all , we have

(11)

If , this implies that .

(12)

Replacing in (12), we get

, for all

If , then , the centralizer of in ,

and hence . On the other hand if ,

then by remark (1) ,we get .

Theorem (3): Let I be a non zero right ideal of a prime ring and f is a left generalized

derivation on with a non zero left derivation on . If f is commuting on , then is

commutative.

Proof: Let f is a commuting on , then for all ,

we have

(13)

Replacing by in (13), we get

(14)

Substituing in (14) and using (13), then we get

[

Open Journal of Applied & Theoretical Mathematics (OJATM)

Vol. 2, No. 4, December 2016, pp. 134~139

ISSN: 2455-7102

Journal homepage:

http://ojal.us/ojatm/

138

, for all .

(15)

Replacing by in (15) and using (13) we get,

, for all

(16)

Replacing in (16), and using (16) , we get

, for all , and

, for all , and . (17)

Replacing by in (17), and using (17), we get

for all , and

Since is a prime ring, and , then , for all , and .

Therefore and so , which implies that is commutative

and by Lemma (3), is commutative.

Theorem (4): Let R be a prime ring and I be a right ideal of such that .

Let f be left generalized derivations on with a non zero left derivation on . If f is

commuting on I, then is commutative.

Proof: Let we take , since f is a commuting on I then the proof is complete.

Now , by equation (11) ,we have

, for all

we replace by ,where , we get

Open Journal of Applied & Theoretical Mathematics (OJATM)

Vol. 2, No. 4, December 2016, pp. 134~139

ISSN: 2455-7102

Journal homepage:

http://ojal.us/ojatm/

139

, for all , and

(18)

By using lemma (1) in (18),we get , and since f is centralizing on I.

We get , for all for all , and

(19)

By using remark (1) in (19), we get

, for all and hence by theorem (1), is commutative.

REFERENCES

[1] Afrah Mohammad Ibraheem: Right ideals and generalized reverse derivations on

prime

rings, American Journal of Computational and Applied

Mathematics,6(4),(2016),162-

164.

[2] Ali.A and shah.T: Centralizing and commuting generalized derivation on prime rings,

Matematiqki Vesnik, 60 (2008),1-2.

[3] Bresar.M: On the distance of the composition of two derivations to the generalized

derivations, Glasgow Math,93(1991),89-93.

[4] Bresar.M: Centralizing mappings and derivations in prime rings, J.Algebra

156,(1993),

385- 394.

[5] Golbasi.O: On left ideals of prime rings with generalized derivations, Hacttepe

Journal of

Mathematics and statistics,Vol.34,(2005),27-32.

[6] Hvala.B: Generalized derivations in prime rings, Comm.Algebra 26(4),(1998),1147-

1166.

[7] Jaya Subba Reddy.C and Mallikarjuna Rao.S: Right ideals of prime rings with left

Generalized derivations, International Journal of Mathematics Trends and

Technology, 25(1),(2015),47-54.

[8] Jaya Subba Reddy.C, Mallikarjuna Rao.S and Vijaya Kumar.V: Centralizing and

Commuting left generalized derivations on prime rings, Bulletin of Mathematical

Science and Applications,Vol.11,(2015),1-3.

[9] Mayne.J.H: Centralizing mappings of prime rings, Cand. Math. Bull.,27(1),

(1984),122-126.

[10] Posner E.C: Derivations in prime rings, Proc.Amer.math.Soc.8,(1957),1093-1100.

ResearchGate has not been able to resolve any citations for this publication.

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Derivations in Prime Rings

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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.

Afrah Mohammad Ibraheem: Right ideals and generalized reverse derivations on prime rings

Jan 2016
162-164

Afrah Mohammad Ibraheem: Right ideals and generalized reverse derivations on prime rings, American Journal of Computational and Applied Mathematics,6(4),(2016),162-164.

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