Remarks on Generalized Derivations in Prime and Semiprime Rings

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Volume 2010, Article ID 646587,6pages doi:10.1155/2010/646587

Research Article

Remarks on Generalized Derivations in

Prime and Semiprime Rings

Basudeb Dhara

Department of Mathematics, Belda College, Belda, Paschim Medinipur 721424, India

Correspondence should be addressed to Basudeb Dhara,basu dhara@yahoo.com

Received 15 August 2010; Accepted 28 November 2010

Academic Editor: Hans Keiding

Copyrightq2010 Basudeb Dhara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

LetRbe a ring with centerZandI a nonzero ideal ofR. An additive mappingF : RRis called a generalized derivation ofRif there exists a derivation d : RRsuch thatFxy Fxyxdyfor allx, yR. In the present paper, we prove that ifFx, y ±x, yfor all x, yIorFxy ±xyfor allx, yI, then the semiprime ringRmust contains a nonzero central ideal, provideddI/0. In caseRis prime ring,Rmust be commutative, providedd /0. The casesiFx, y±x, yZandiiFxy±xyZfor allx, yIare also studied.

1. Introduction

LetRbe an associative ring. The center ofRis denoted byZ. Forx, yR, the symbolx, y

will denote the commutatorxyyxand the symbolxywill denote the anticommutator

xyyx. We will make extensive use of basic commutator identitiesxy, z x, zyxy, z,

x, yz x, yzyx, z. An additive mappingdfromRtoRis called a derivation ofRif

dxy dxyxdyholds for allx, yR. An additive mappingg fromRtoRis called a generalized derivation of Rif there exists a derivationd fromR toR such thatgxy gxyxdyholds for allx, yR. Obviously, every derivation is a generalized derivation ofR. Thus, generalized derivation covers both the concept of derivation and left multiplier mapping. A mappingFfromRtoRis called centralizing onSwhereSR, ifFx, xZ

for allxS.

Over the last several years, a number of authors studied the commutativity in prime and semiprime rings admitting derivations and generalized derivations. In1, Daif and Bell

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is commutative. Recently, Quadri et al.2generalized this result replacing derivationdwith a generalized derivation in a prime ringR. More precisely, they proved the following.

Let R be a prime ring andI a nonzero ideal of R. If R admits a generalized derivation F associated with a nonzero derivationdsuch that any one of the following holds: (i) Fx, y x, y for allx, yI, (ii) Fx, yx, yfor allx, yI, (iii) Fxy xyfor allx, yI; (iv) Fxyxyfor allx, yI, thenRis commutative.

In the present paper, we study all these cases in semiprime ring.

2. Main Results

We recall some known results on prime and semiprime rings.

Lemma 2.1see3, Lemma 1.1.5or1, Lemma 2. aIfRis a semiprime ring, the center of a nonzero one-sided ideal is contained in the center ofR, in particular, any commutative one-sided ideal is contained in the center ofR.

bIfRis a prime ring with a nonzero central ideal, thenRmust be commutative.

Lemma 2.2see1, Lemma 1. LetRbe a semiprime ring andIa nonzero ideal ofR. IfzRand zcentralizesI, I, thenzcentralizesI.

Lemma 2.3see4, Theorem 3. LetRbe a semiprime ring andUa nonzero left ideal ofR. IfR admits a derivationdwhich is nonzero onUand centralizing onU, thenRcontains a nonzero central ideal.

Now we begin with the theorem.

Theorem 2.4. LetRbe a semiprime ring,Ia nonzero ideal ofRandFa generalized derivation ofR associated with a derivationdofRsuch thatdI/0. IfFx, y ±x, yfor allx, yI, thenR contains a nonzero central ideal.

Proof. By our assumption, we have that

Fx, y±x, y 2.1

for allx, yI. IfFI 0, then we find thatx, y 0 for allx, yI, that is,Iis commutative. Then, byLemma 2.1,IZand thus we obtain our conclusion.

Next assume thatFI/0. Puttingyyxin2.1, we get that

Fx, yx±x, yx. 2.2

SinceFis a generalized derivation ofRassociated with a derivationdofR,2.2gives

Fx, yxx, ydx ±x, yx. 2.3

Using2.1, it reduces to

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for allx, yI. Now puttingydxyin2.4, we get

0x, dxydx dxx, ydx x, dxydx. 2.5

Using2.4, it gives

0 x, dxydx 2.6

for allx, yI. Now we putyyxin2.6and obtain that

0 x, dxyxdx 2.7

for allx, yI. Right multiplying2.6byxand then subtracting from2.7, we get

0 x, dxyx, dx 2.8

for allx, yI. This implies for allxI thatx, dxI2 0 and sox, dxI 0, forcing x, dxI∩AnnI 0. Then byLemma 2.3,Rcontains a nonzero central ideal.

Corollary 2.5. LetRbe a prime ring,I a nonzero ideal ofRandFa generalized derivation ofR. If Fx, y ±x, yfor allx, yI, thenRis commutative orFx ±xfor allxI.

Proof. Letdbe the associated derivation ofF. ByTheorem 2.4, we conclude that eitherdI

0 orRis commutative. Assume thatRis not commutative. ThendI 0. SinceRis a prime ring,dI 0 impliesdR 0 and henceFxy Fxyfor allx, yR. SetGx Fxx

for allxR. ThenGxy Gxyfor allxR. Now, our assumptionFx, y ±x, y

givesFxyFyx ±xyyx, that is,GxyGyx 0 for allx, yI. Thus using

Gxy Gyx, we haveGxyz Gyxz Gxzy Gxzy, that is,Gxy, z 0 for allx, y, zI. Thus 0 GII, I GIRI, I GIRI, I. SinceRis prime, this implies

GI 0 orI is commutative. ByLemma 2.1,Icommutative implies thatRis commutative, a contradiction. ThusGI 0 which givesGx Fxx0 for allxI.

Theorem 2.6. LetRbe a semiprime ring,Ia nonzero ideal ofRandFa generalized derivation ofR associated with a derivationdofRsuch thatdI/0. IfFxy ±xyfor allx, yI, thenR contains a nonzero central ideal.

Proof. IfFI 0, then by our assumption we have thatxy 0, that is,xyyx 0 for allx, yI. This implies thatxyzyzx −yzx yxz yxzxyzfor all

x, y, zIand so 2I3 0, forcing 2I0. Therefore, for allx, yI,xyyx0 givesxyyx,

that is,Iis commutative. Then byLemma 2.1,IZand thus we obtain our conclusion.

Next assume thatFI/0. Then for anyx, yI, we have

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SinceFis a generalized derivation associated with a derivationd, above expression yields

FxyxdyFyxydx ±xyyx. 2.10

Puttingyyxin2.10, we have

FxyxxdyxydxFyxydxxyxdx ±xyxyx2. 2.11

Right multiplying2.10byxand then subtracting from2.11, we get

xydx yxdx 0 2.12

for allx, yI. Replacingywithdxyin2.12and then again using2.12we find that

x, dxydx 0. 2.13

Again replacingywithyxin2.13and then using2.13we obtain

x, dxyx, dx 0 2.14

for allx, yI, which is the same identity as2.8in the proof ofTheorem 2.4. Thus by the same argument as in the proof ofTheorem 2.4, we conclude thatRcontains a nonzero central ideal.

Corollary 2.7. LetRbe a prime ring,I a nonzero ideal ofRandFa generalized derivation ofR. If Fxy ±xyfor allx, yI, thenRis commutative orFx ±xfor allxI.

Proof. Letdbe the associated derivation ofF. ByTheorem 2.6, we conclude that eitherdI

0 orRis commutative. IfRis not commutative, thendI 0. SinceRis a prime ring,dI

0 implies dR 0 and hence Fxy Fxy for all x, yR. SetGx Fxx for all xR. ThenGxy Gxy for allxR. Now, our assumptionFxy ±xy

givesFxyFyx ±xyyx, that is,GxyGyx 0 for allx, yI. Thus using

Gxy−Gyx, we haveGxyz−GyxzGxzyGxzy, that is,Gxy, z 0 for allx, y, zI. Thus 0 GII, I GIRI, I GIRI, I. SinceRis prime, this implies

GI 0 orI is commutative. ByLemma 2.1,Icommutative implies thatRis commutative, a contradiction. Therefore,GI 0 and henceGx Fxx0 for allxI.

Theorem 2.8. Let R be a semiprime ring with center Z /{0}, I a nonzero ideal of R and F a generalized derivation ofR associated with a derivationdof R. If Fx, y±x, yZ for all x, yI, thenIdZZ.

Proof. We have

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for allx, yI. SinceZ /{0}, we may choose 0/zZ. ThenyzIfor anyyI. Now we replaceywithyzin2.15and then we get

Fx, yz±x, yzFx, yzx, ydz±x, yz

Fx, y±x, y zx, ydzZ. 2.16

By 2.15, we havex, ydzZfor allx, yI. SincedzZ, this gives that for any

rR,r,x, ydz 0 which impliesrdz,x, y 0 for allx, yI. ByLemma 2.2, rdz, x 0 for allxI. SincedzZ, this givesr, xdz 0 for allrRand for all

xI. Thus,xdzZ, that is,IdZZ.

Corollary 2.9. LetRbe a prime ring with centerZ /{0},Ia nonzero ideal ofRandFa generalized derivation ofRassociated with a derivationd. IfdZ/{0}andFx, y±x, yZfor allx, yI, thenRis commutative.

Proof. SincedZZandZcontains no nonzero elements which are zero divisors, we have fromTheorem 2.8thatIZ. Then byLemma 2.1b, we obtain our conclusion.

Theorem 2.10. Let R be a semiprime ring with center Z /{0}, I a nonzero ideal of R and F a generalized derivation ofR associated with a derivationd ofR. If Fxy±xyZ for all x, yI, thenIdZZ.

Proof. We have

Fxy±xyZ 2.17

for allx, yI. SinceZ /{0}, we choose 0/zZ. ThenyzIfor anyyI. Now we replace

ywithyzin2.17and then we get

Fxyz±xyzFxyzxydz±xyz

Fxy±xy zxydzZ. 2.18

By2.17, we havex◦ydzZthat isxyyxdzZfor allx, yI. Now puttingyyr

andxrx,rR, respectively, we obtain thatxyryrxdzZandrxyyrxdzZ. Subtracting these two results yieldsxydz, rZ for allx, yI and for allrR. This gives

xydz, r, s0 2.19

for allx, yI and for allr, sR. We know the Jacobian identityx, y, z y, z, x

z, x, y 0 for anyx, y, zR. Using this identity, it follows that

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By using2.19, it reduces to

r, s, xydz0 2.21

for allr, sR and for all x, yI. ByLemma 2.2, this implies thatxydz, r 0, that is,I2dz, R 0. ThusI, I, Idz 0 and then again byLemma 2.2,I, Idz 0. This

yields 0 IR, Idz IR, Idzwhich impliesIdzZ, sinceR, IdzI∩AnnI 0. Sincezis any nonzero element inZ, we conclude thatIdZZ.

Acknowledgment

The author would like to thank the referees for providing very helpful comments and suggestions.

References

1 M. N. Daif and H. E. Bell, “Remarks on derivations on semiprime rings,” International Journal of Mathematics and Mathematical Sciences, vol. 15, no. 1, pp. 205–206, 1992.

2 M. A. Quadri, M. S. Khan, and N. Rehman, “Generalized derivations and commutativity of prime rings,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 9, pp. 1393–1396, 2003.

3 I. N. Herstein, Rings with Involution. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill, USA, 1976.

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