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.

Let*R*be a ring with center*Z*and*I* a nonzero ideal of*R. An additive mappingF* : *R* → *R*is
called a generalized derivation of*R*if there exists a derivation *d* : *R* → *R*such that*Fxy*
*Fxyxdy*for all*x, y* ∈ *R. In the present paper, we prove that ifFx, y* ±*x, y*for all
*x, y*∈*I*or*Fx*◦*y* ±*x*◦*y*for all*x, y*∈*I*, then the semiprime ring*R*must contains a nonzero
central ideal, provided*dI/*0. In case*R*is prime ring,*R*must be commutative, provided*d /*0.
The casesi*Fx, y*±*x, y*∈*Z*andii*Fx*◦*y*±*x*◦*y*∈*Z*for all*x, y*∈*I*are also studied.

**1. Introduction**

Let*R*be an associative ring. The center of*R*is denoted by*Z*. For*x, y*∈*R*, the symbol*x, y*

will denote the commutator*xy*−*yx*and the symbol*x*◦*y*will denote the anticommutator

*xyyx*. We will make extensive use of basic commutator identities*xy, z* *x, zyxy, z*,

*x, yz* *x, yzyx, z*. An additive mapping*d*from*R*to*R*is called a derivation of*R*if

*dxy* *dxyxdy*holds for all*x, y* ∈ *R*. An additive mapping*g* from*R*to*R*is called
a generalized derivation of *R*if there exists a derivation*d* from*R* to*R* such that*gxy*
*gxyxdy*holds for all*x, y* ∈*R*. Obviously, every derivation is a generalized derivation
of*R*. Thus, generalized derivation covers both the concept of derivation and left multiplier
mapping. A mapping*F*from*R*to*R*is called centralizing on*S*where*S*⊆*R*, if*Fx, x*∈*Z*

for all*x*∈*S*.

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

is commutative. Recently, Quadri et al.2generalized this result replacing derivation*d*with
a generalized derivation in a prime ring*R*. 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, y*∈*I, (ii)* *Fx, y* −*x, yfor allx, y*∈*I, (iii)* *Fx*◦*y* *x*◦*yfor allx, y*∈*I; (iv)*
*Fx*◦*y* −*x*◦*yfor allx, y*∈*I, 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 2.1**see3, Lemma 1.1.5or1, Lemma 2. a*IfRis 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.*

b*IfRis a prime ring with a nonzero central ideal, thenRmust be commutative.*

**Lemma 2.2**see1, Lemma 1. Let*Rbe a semiprime ring andIa nonzero ideal ofR. Ifz*∈*Rand*
*zcentralizesI, I, thenzcentralizesI.*

**Lemma 2.3**see4, Theorem 3. Let*Rbe 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. Let**Rbe a semiprime ring,Ia nonzero ideal ofRandFa generalized derivation ofR*associated with a derivationdofRsuch thatdI/0. IfFx, y* ±*x, yfor allx, y*∈*I, thenR*
*contains a nonzero central ideal.*

*Proof. By our assumption, we have that*

*Fx, y*±*x, y* 2.1

for all*x, y*∈*I*. If*FI* 0, then we find that*x, y* 0 for all*x, y*∈*I*, that is,*I*is commutative.
Then, byLemma 2.1,*I*⊆*Z*and thus we obtain our conclusion.

Next assume that*FI/*0. Putting*yyx*in2.1, we get that

*Fx, yx*±*x, yx.* 2.2

Since*F*is a generalized derivation of*R*associated with a derivation*d*of*R*,2.2gives

*Fx, yxx, ydx* ±*x, yx.* 2.3

Using2.1, it reduces to

for all*x, y*∈*I*. Now putting*ydxy*in2.4, we get

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

Using2.4, it gives

0 *x, dxydx* 2.6

for all*x, y*∈*I*. Now we put*yyx*in2.6and obtain that

0 *x, dxyxdx* 2.7

for all*x, y*∈*I*. Right multiplying2.6by*x*and then subtracting from2.7, we get

0 *x, dxyx, dx* 2.8

for all*x, y*∈*I*. This implies for all*x*∈*I* that*x, dxI*2 0 and so*x, dxI* 0, forcing
*x, dx*∈*I*∩Ann*I* 0. Then byLemma 2.3,*R*contains a nonzero central ideal.

**Corollary 2.5. Let**Rbe a prime ring,I*a nonzero ideal ofRandFa generalized derivation ofR. If*
*Fx, y* ±*x, yfor allx, y*∈*I, thenRis commutative orFx* ±x*for allx*∈*I.*

*Proof. Letd*be the associated derivation of*F*. ByTheorem 2.4, we conclude that either*dI*

0 or*R*is commutative. Assume that*R*is not commutative. Then*dI* 0. Since*R*is a prime
ring,*dI* 0 implies*dR* 0 and hence*Fxy* *Fxy*for all*x, y*∈*R*. Set*Gx* *Fx*∓*x*

for all*x* ∈ *R*. Then*Gxy* *Gxy*for all*x* ∈*R*. Now, our assumption*Fx, y* ±*x, y*

gives*Fxy*−*Fyx* ±*xy*−*yx*, that is,*Gxy*−*Gyx* 0 for all*x, y* ∈ *I*. Thus using

*Gxy* *Gyx*, we have*Gxyz* *Gyxz* *Gxzy* *Gxzy*, that is,*Gxy, z* 0 for
all*x, y, z*∈*I*. Thus 0 *GII, I* *GIRI, I* *GIRI, I*. Since*R*is prime, this implies

*GI* 0 or*I* is commutative. ByLemma 2.1,*I*commutative implies that*R*is commutative,
a contradiction. Thus*GI* 0 which gives*Gx* *Fx*∓*x*0 for all*x*∈*I*.

**Theorem 2.6. Let**Rbe a semiprime ring,Ia nonzero ideal ofRandFa generalized derivation ofR*associated with a derivationdofRsuch thatdI/0. IfFx*◦*y* ±*x*◦*yfor allx, y*∈*I, thenR*
*contains a nonzero central ideal.*

*Proof. IfFI* 0, then by our assumption we have that*x*◦*y* 0, that is,*xyyx* 0 for
all*x, y* ∈ *I*. This implies that*xyz* −*yzx* −y*zx* *yxz* *yxz* −*xyz*for all

*x, y, z*∈*I*and so 2*I*3 _{}_{0, forcing 2}_{I}_{}_{0. Therefore, for all}_{x, y}_{∈}_{I}_{,}_{xy}_{}_{yx}_{}_{0 gives}_{xy}_{}_{yx}_{,}

that is,*I*is commutative. Then byLemma 2.1,*I*⊆*Z*and thus we obtain our conclusion.

Next assume that*FI/*0. Then for any*x, y*∈*I*, we have

Since*F*is a generalized derivation associated with a derivation*d*, above expression yields

*FxyxdyFyxydx* ±*xyyx.* 2.10

Putting*yyx*in2.10, we have

*FxyxxdyxydxFyxydxxyxdx* ±*xyxyx*2*.* 2.11

Right multiplying2.10by*x*and then subtracting from2.11, we get

*xydx* *yxdx* 0 2.12

for all*x, y*∈*I*. Replacing*y*with*dxy*in2.12and then again using2.12we find that

*x, dxydx* 0*.* 2.13

Again replacing*y*with*yx*in2.13and then using2.13we obtain

*x, dxyx, dx* 0 2.14

for all*x, y* ∈*I*, 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 that*R*contains a nonzero central
ideal.

**Corollary 2.7. Let**Rbe a prime ring,I*a nonzero ideal ofRandFa generalized derivation ofR. If*
*Fx*◦*y* ±*x*◦*yfor allx, y*∈*I, thenRis commutative orFx* ±x*for allx*∈*I.*

*Proof. Letd*be the associated derivation of*F*. ByTheorem 2.6, we conclude that either*dI*

0 or*R*is commutative. If*R*is not commutative, then*dI* 0. Since*R*is a prime ring,*dI*

0 implies *dR* 0 and hence *Fxy* *Fxy* for all *x, y* ∈ *R*. Set*Gx* *Fx*∓*x* for
all *x* ∈ *R*. Then*Gxy* *Gxy* for all*x* ∈ *R*. Now, our assumption*Fx*◦*y* ±*x*◦*y*

gives*FxyFyx* ±*xyyx*, that is,*GxyGyx* 0 for all*x, y* ∈ *I*. Thus using

*Gxy*−G*yx*, we have*Gxyz*−G*yxzGxzyGxzy*, that is,*Gxy, z* 0 for
all*x, y, z*∈*I*. Thus 0 *GII, I* *GIRI, I* *GIRI, I*. Since*R*is prime, this implies

*GI* 0 or*I* is commutative. ByLemma 2.1,*I*commutative implies that*R*is commutative,
a contradiction. Therefore,*GI* 0 and hence*Gx* *Fx*∓*x*0 for all*x*∈*I*.

**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, y* ∈ *Z* *for all*
*x, y*∈*I, thenIdZ*⊆*Z.*

*Proof. We have*

for all*x, y*∈*I*. Since*Z /*{0}, we may choose 0*/z*∈*Z*. Then*yz* ∈*I*for any*y* ∈*I*. Now we
replace*y*with*yz*in2.15and then we get

*Fx, yz*±*x, yzFx, yzx, ydz*±*x, yz*

*Fx, y*±*x, y* *zx, ydz*∈*Z.* 2.16

By 2.15, we have*x, ydz* ∈ *Z*for all*x, y* ∈ *I*. Since*dz* ∈ *Z*, this gives that for any

*r* ∈ *R*,*r,x, ydz* 0 which implies*rdz,x, y* 0 for all*x, y* ∈ *I*. ByLemma 2.2,
*rdz, x* 0 for all*x*∈*I*. Since*dz* ∈*Z*, this gives*r, xdz* 0 for all*r* ∈*R*and for all

*x*∈*I*. Thus,*xdz*∈*Z*, that is,*IdZ*⊆*Z*.

* 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, y*∈

*Zfor allx, y*∈

*I,*

*thenRis commutative.*

*Proof. SincedZ*⊆*Z*and*Z*contains no nonzero elements which are zero divisors, we have
fromTheorem 2.8that*I*⊆*Z*. 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* *Fx*◦*y*±*x*◦*y* ∈ *Z* *for all*
*x, y*∈*I, thenIdZ*⊆*Z.*

*Proof. We have*

*Fx*◦*y*±*x*◦*y*∈*Z* 2.17

for all*x, y*∈*I*. Since*Z /*{0}, we choose 0*/z*∈*Z*. Then*yz*∈*I*for any*y*∈*I*. Now we replace

*y*with*yz*in2.17and then we get

*Fx*◦*yz*±*x*◦*yzFx*◦*yzx*◦*ydz*±*x*◦*yz*

*Fx*◦*y*±*x*◦*y* *zx*◦*ydz*∈*Z.* 2.18

By2.17, we have*x◦ydz*∈*Z*that is*xyyxdz*∈*Z*for all*x, y*∈*I*. Now putting*yyr*

and*xrx*,*r*∈*R*, respectively, we obtain that*xyryrxdz*∈*Z*and*rxyyrxdz*∈*Z*.
Subtracting these two results yields*xydz, r* ∈ *Z* for all*x, y* ∈ *I* and for all*r* ∈ *R*. This
gives

*xydz, r, s*0 2.19

for all*x, y* ∈ *I* and for all*r, s* ∈ *R*. We know the Jacobian identity*x, y, z* *y, z, x*

*z, x, y* 0 for any*x, y, z*∈*R*. Using this identity, it follows that

By using2.19, it reduces to

*r, s, xydz*0 2.21

for all*r, s* ∈ *R* and for all *x, y* ∈ *I*. ByLemma 2.2, this implies that*xydz, r* 0, that
is,*I*2_{d}_{}_{z}_{}_{, R}_{0. Thus}_{}_{I, I}_{}_{, Id}_{}_{z}_{0 and then again by}_{Lemma 2.2}_{,}_{}_{I, Id}_{}_{z}_{0. This}

yields 0 *IR, Idz* *IR, Idz*which implies*Idz*⊆*Z*, since*R, Idz*⊆*I∩*Ann*I* 0.
Since*z*is any nonzero element in*Z*, we conclude that*IdZ*⊆*Z*.

**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.