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 [email protected] 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 : R → Ris called a generalized derivation ofRif there exists a derivation d : R → Rsuch thatFxy Fxyxdyfor allx, y ∈ R. In the present paper, we prove that ifFx, y ±x, yfor all x, y∈IorFx◦y ±x◦yfor allx, y∈I, then the semiprime ringRmust contains a nonzero central ideal, provideddI/0. In caseRis prime ring,Rmust be commutative, providedd /0.
The casesiFx, y±x, y∈ZandiiFx◦y±x◦y∈Zfor allx, y∈Iare also studied.
1. Introduction
LetRbe an associative ring. The center ofRis denoted byZ. Forx, y∈R, the symbolx, y will denote the commutatorxy−yxand the symbolx◦ywill 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, y ∈ R. An additive mappingg fromRtoRis called a generalized derivation of Rif there exists a derivationd fromR toR such thatgxy gxyxdyholds for allx, y ∈R. 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 onSwhereS⊆R, ifFx, x∈Z for allx∈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 proved that ifRis a semiprime ring with a nonzero idealK anddis a derivation ofRsuch thatdx, y ±x, yfor allx, y∈K, thenKis central ideal. In particular, ifKR, thenR
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, 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 2.1see3, 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. Ifz∈Rand 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, y∈I, thenR contains a nonzero central ideal.
Proof. By our assumption, we have that F
x, y ±
x, y
2.1 for allx, y∈I. IfFI 0, then we find thatx, y 0 for allx, y∈I, that is,Iis commutative.
Then, byLemma 2.1,I⊆Zand thus we obtain our conclusion.
Next assume thatFI/0. Puttingyyxin2.1, we get that F
x, y x
± x, y
x. 2.2
SinceFis a generalized derivation ofRassociated with a derivationdofR,2.2gives F
x, y x
x, y
dx ± x, y
x. 2.3
Using2.1, it reduces to
x, y
dx 0 2.4
for allx, y∈I. Now puttingydxyin2.4, we get
0
x, dxy
dx dx x, y
dx x, dxydx. 2.5
Using2.4, it gives
0 x, dxydx 2.6
for allx, y∈I. Now we putyyxin2.6and obtain that
0 x, dxyxdx 2.7
for allx, y∈I. Right multiplying2.6byxand then subtracting from2.7, we get
0 x, dxyx, dx 2.8
for allx, y∈I. This implies for allx∈I thatx, dxI2 0 and sox, dxI 0, forcing x, dx∈I∩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, y∈I, thenRis commutative orFx ±xfor allx∈I.
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, y∈R. SetGx Fx∓x for allx ∈ R. ThenGxy Gxyfor allx ∈R. Now, our assumptionFx, y ±x, y givesFxy−Fyx ±xy−yx, that is,Gxy−Gyx 0 for allx, y ∈ I. Thus using Gxy Gyx, we haveGxyz Gyxz Gxzy Gxzy, that is,Gxy, z 0 for allx, y, z∈I. 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 Fx∓x0 for allx∈I.
Theorem 2.6. LetRbe 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 thatx◦y 0, that is,xyyx 0 for allx, y ∈ I. This implies thatxyz −yzx −yzx yxz yxz −xyzfor all x, y, z∈Iand so 2I3 0, forcing 2I0. Therefore, for allx, y∈I,xyyx0 givesxyyx, that is,Iis commutative. Then byLemma 2.1,I⊆Zand thus we obtain our conclusion.
Next assume thatFI/0. Then for anyx, y∈I, we have
F
xyyx ±
xyyx
. 2.9
SinceFis a generalized derivation associated with a derivationd, above expression yields Fxyxd
y F
y
xydx ±
xyyx
. 2.10
Puttingyyxin2.10, we have Fxyxx
d y
xydx
F y
xydx
xyxdx ±
xyxyx2
. 2.11
Right multiplying2.10byxand then subtracting from2.11, we get
xydx yxdx 0 2.12
for allx, y∈I. 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, 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 thatRcontains a nonzero central ideal.
Corollary 2.7. LetRbe a prime ring,I a nonzero ideal ofRandFa generalized derivation ofR. If Fx◦y ±x◦yfor allx, y∈I, thenRis commutative orFx ±xfor allx∈I.
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, y ∈ R. SetGx Fx∓x for all x ∈ R. ThenGxy Gxy for allx ∈ R. Now, our assumptionFx◦y ±x◦y givesFxyFyx ±xyyx, that is,GxyGyx 0 for allx, y ∈ I. Thus using Gxy−Gyx, we haveGxyz−GyxzGxzyGxzy, that is,Gxy, z 0 for allx, y, z∈I. 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 Fx∓x0 for allx∈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
F x, y
± x, y
∈Z 2.15
for allx, y∈I. SinceZ /{0}, we may choose 0/z∈Z. Thenyz ∈Ifor anyy ∈I. Now we replaceywithyzin2.15and then we get
F x, y
z
± x, y
zF x, y
z x, y
dz± x, y
z
F x, y
±
x, y z x, y
dz∈Z. 2.16
By 2.15, we havex, ydz ∈ Zfor allx, y ∈ I. Sincedz ∈ Z, this gives that for any r ∈ R,r,x, ydz 0 which impliesrdz,x, y 0 for allx, y ∈ I. ByLemma 2.2, rdz, x 0 for allx∈I. Sincedz ∈Z, this givesr, xdz 0 for allr ∈Rand 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⊆ZandZcontains no nonzero elements which are zero divisors, we have fromTheorem 2.8thatI⊆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
F x◦y
± x◦y
∈Z 2.17
for allx, y∈I. SinceZ /{0}, we choose 0/z∈Z. Thenyz∈Ifor anyy∈I. Now we replace ywithyzin2.17and then we get
F x◦y
z
± x◦y
zF x◦y
z x◦y
dz± x◦y
z
F x◦y
±
x◦y z x◦y
dz∈Z. 2.18
By2.17, we havex◦ydz∈Zthat isxyyxdz∈Zfor allx, y∈I. Now puttingyyr andxrx,r∈R, respectively, we obtain thatxyryrxdz∈Zandrxyyrxdz∈Z.
Subtracting these two results yieldsxydz, r ∈ Z for allx, y ∈ I and for allr ∈ R. This gives
xydz, r , s
0 2.19
for allx, y ∈ I and for allr, s ∈ R. We know the Jacobian identityx, y, z y, z, x z, x, y 0 for anyx, y, z∈R. Using this identity, it follows that
0
xydz, r , s
−
r, s, xydz
−
s, xydz , r
. 2.20
By using2.19, it reduces to
r, s, xydz
0 2.21
for allr, s ∈ R and for all x, y ∈ I. 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 impliesIdz⊆Z, sinceR, Idz⊆I∩AnnI 0.
Sincezis any nonzero element inZ, we conclude thatIdZ⊆Z.
Acknowledgment
The author would like to thank the referees for providing very helpful comments and suggestions.
References
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3 I. N. Herstein, Rings with Involution. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill, USA, 1976.
4 H. E. Bell and W. S. Martindale III, “Centralizing mappings of semiprime rings,” Canadian Mathematical Bulletin, vol. 30, no. 1, pp. 92–101, 1987.
