Generalized Derivations of Prime Rings

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International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 85612,6pages

doi:10.1155/2007/85612

Research Article

Generalized Derivations of Prime Rings

Huang Shuliang

Received 10 January 2007; Revised 8 May 2007; Accepted 19 June 2007 Recommended by Akbar Rhemtulla

Let R be an associative prime ring, U a Lie ideal such that u2U for all uU. An additive functionF:R→Ris called a generalized derivation if there exists a derivation d:R→Rsuch thatF(xy)=F(x)y+xd(y) holds for allx,y∈R. In this paper, we prove thatd=0 orU⊆Z(R) if any one of the following conditions holds: (1)d(x)◦F(y)=0, (2) [d(x),F(y)=0], (3) either d(x)◦F(y)=x◦yord(x)◦F(y) +x◦y=0, (4) either d(x)◦F(y)=[x,y] or d(x)◦F(y) + [x,y]=0, (5) either d(x)◦F(y)−xy∈Z(R) or d(x)◦F(y) +xy∈Z(R), (6) either [d(x),F(y)]=[x,y] or [d(x),F(y)] + [x,y]=0, (7) either [d(x),F(y)]=x◦yor [d(x),F(y)] +x◦y=0 for allx,y∈U.

Copyright © 2007 Huang Shuliang. 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.

1. Introduction

LetRbe an associative ring with centerZ(R). For anyx,y∈R, the symbol [x,y] stands for the commutator xy−yxand denote by x◦y the anticommutatorxy+yx. Given two subsetsAandBofR, [A,B] will denote the additive subgroup ofRgenerated by all elements of the form [a,b], wherea∈A,b∈B. For a nonempty subsetSofR, we put CR(S)= {x∈R|[x,s]=0 for alls∈S}. Recall thatRis prime ifaRb=0 impliesa=0 orb=0. An additive mapd:R→Ris called a derivation ifd(xy)=d(x)y+xd(y) holds for allx,y∈R. An additive functionF:R→Ris called a generalized derivation if there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y) holds for allx,y∈R. Many analysts have studied generalized derivation in the context of algebras on certain normed spaces (see [1] for reference). An additive subgroupUofRis said to be a Lie ideal ofRif [u,r]∈Ufor allu∈Uandr∈R.

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properties:d(x)◦F(y)=0, [d(x),F(y)]=0,d(x)◦F(y)=x◦y,d(x)◦F(y) +x◦y=0, d(x)◦F(y)−xy∈Z(R), d(x)◦F(y) +xy∈Z(R), [d(x),F(y)]=[x,y], [d(x),F(y)] + [x,y]=0 for allx,y∈I, whereIis a nonzero ideal ofR.

In [3], Bergen et al. investigate the relationship between the derivations and Lie ideals of a prime ring, and obtain some useful results. In [4], P. H. Lee and T. K. Lee get six sufficient conditions of central Lie ideal, which extend some results of commutativity on a prime ring. Motivated by the above, in this paper, we extend M. Ashraf ’s results to a Lie ideal of a prime ring. Throughout this paper,Rwill be a prime ring andU will always denote a Lie ideal ofR.

2. Preliminaries

We begin with the following known results which will be used extensively to prove our theorems.

Lemma 2.1. IfUis a Lie ideal of R such thatu2Ufor alluU, then 2uvUfor allu, v∈U.

Proof. For allw,u,v∈U,

uv+vu=(u+v)2−u2−v2∈U. (2.1)

On the other hand,

uv−vu∈U. (2.2)

Adding two expressions, we have 2uv∈Ufor allu,v∈U. Lemma 2.2 [3]. IfUZ(R) is a Lie ideal ofR, thenCR(U)=Z(R).

Lemma 2.3 [3]. IfUis a Lie ideal ofR, thenCR([U,U])=CR(U).

Lemma 2.4. SetV= {u∈U|d(u)∈U}. IfUZ(R), thenVZ(R).

Proof. Assume thatV ⊆Z(R). Since [U,U]⊆U andd([U,U])⊆U, we have [U,U]⊆

V⊆Z(R). HenceCR([U,U])=R. FromLemma 2.2,CR(U)=Z(R). But byLemma 2.3, CR([U,U])=CR(U). That is,R=Z(R), a contradiction. Lemma 2.5 [3]. IfUZ(R) is a Lie ideal ofRand ifaUb=0, thena=0 orb=0.

Lemma 2.6. A group can not be a union of two of its proper subgroups.

Lemma 2.7 [4]. Letd=0 be a derivation of R such that [u,d(u)]∈Z(R) for allu∈U. ThenU⊆Z(R).

Lemma 2.8 [4]. Let dandδbe nonzero derivations ofRsuch thatdδ(U)⊆Z(R). Then U⊆Z(R).

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Lemma 2.10 [3]. Ifd=0 is a derivation ofR, and ifUis a Lie ideal ofRsuch thatd(U)⊆

Z(R), thenU⊆Z(R).

3. The proof of main theorems

Theorem 3.1. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2U for

allu∈U, andF a generalized derivation associated withd=0. Ifd(x)◦F(y)=0 for all x,y∈U, thenU⊆Z(R)s.

Proof. Assume thatUZ(R), thenVZ(R) byLemma 2.4. Now we haved(x)◦F(y)=

0 for allx,y∈U. Replacingyby 2yz, byLemma 2.1and using charR=2, we getd(x)◦

F(yz)=0 for allx,y,z∈Uand we obtain (d(x)◦y)d(z)−y[d(x),d(z)]+(d(x)◦F(y))z−

F(y)[d(x),z]=0. Now using our hypotheses, the above relation yields (d(x)◦y)d(z)−

y[d(x),d(z)]−F(y)[d(x),z]=0. For anyx∈V, replacezbyd(x) to get

d(x)◦yd2(x)yd(x),d2(x)=0. (3.1)

Now, replaceyby 2zyin (3.1) to get (d(x)(zy))d2(x)zy[d(x),d2(x)]=0. This im-plies that

zd(x)◦yd2(x) +d(x),zyd2(x)zyd(x),d2(x)=0. (3.2)

Combining (3.1) with (3.2), we get [d(x),z]yd2(x)=0 forxVandy,zU. In particu-lar, [d(x),x]yd2(x)=0, and hence [d(x),x]Ud2(x)=0 forxV. Then, either [d(x),x]= 0 ord2(x)=0 by Lemma 2.5. Now let V

1= {x∈V |[d(x),x]=0} and V2= {x∈V | d2(x)=0}. ThenV

1,V2 are both additive subgroups ofV andV1∪V2=V. Thus, ei-therV =V1 orV =V2 byLemma 2.6. IfV=V1, then Lemma 2.7givesV ⊆Z(R), a contradiction. On the other hand, ifV=V2thenV⊆Z(R) byLemma 2.8, again a

con-tradiction.

Theorem 3.2. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2U for

allu∈U, andF a generalized derivation associated withd=0. If [d(x),F(y)]=0 for all x,y∈U, thenU⊆Z(R).

Proof. Assume thatUZ(R), thenVZ(R) byLemma 2.4. By hypotheses, we have

d(x),F(y)=0 (3.3)

for allx,y∈U. Replacingyby 2yzin (3.3) and using charR=2, we get

F(y)d(x),z+yd(x),d(z)+d(x),yd(z)=0 (3.4)

for allx,y,z∈U. For anyx∈V, replacingzby 2zd(x) in (3.4) and using (3.4), we get

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Again, replacingyby 2tyin (3.5) and using (3.5), we get [d(x),t]yzd2(x)=0 for allxV andy,z,t∈U. In particular, [d(x),x]yzd2(x)=0. That is, [d(x),x]UUd2(x)=0. Then, either [d(x),x]=0 orUd2(x)=0 (i.e.,d2(x)=0 ). Now, letV

1= {x∈V|[d(x),x]= 0}andV2= {x∈V |Ud2(x)=0}. ThenV1,V2are both additive subgroups ofV and V1∪V2=V. Thus, eitherV=V1orV=V2byLemma 2.6. IfV=V1, thenLemma 2.7 givesV ⊆Z(R), a contradiction. On the other hand, if V =V2 thenUd2(x)=0, and henced2(x)Ud2(x)=0 for allxV. Thus,Lemma 2.5yields thatd2(x)=0 for allxV. Now, we have obtainedV⊆Z(R) byLemma 2.8, again a contradiction. Theorem 3.3. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andF a generalized derivation associated withd=0. Ifd(x)◦F(y)=x◦yfor all

x,y∈U, thenU⊆Z(R).

Proof. We are givend(x)◦F(y)=x◦yfor allx,y∈U. IfF=0, thenx◦y=0 for all x,y∈U. Replacingyby 2yzand using charR=2, we gety[x,z]=0 for allx,y,z∈U. In particular, [x,z]y[x,z]=0 (i.e., [x,z]U[x,z]=0), and hence [x,z]=0 (i.e., [U,U]=0) byLemma 2.5. ThenLemma 2.9yields that the required result. Therefore, we assume that F=0. For anyx,y∈U, we have

d(x)◦F(y)=x◦y. (3.6)

Replacingyby 2yz, we get

d(x)◦yd(z)−yd(x),d(z)+d(x)◦F(y)z−F(y)d(x),z=(x◦y)z−y[x,z]. (3.7)

Combining (3.6) with (3.7), we get

d(x)◦yd(z)−yd(x),d(z)−F(y)d(x),z+y[x,z]=0 (3.8)

for allx,y,z∈U. For anyx∈V, replacingzbyd(x) in (3.8), we get

d(x)◦yd2(x)yd(x),d2(x)+yx,d(x)=0. (3.9)

Now, replacingyby 2yzin (3.9), we get

zd(x)◦y+d(x),zyd2(x)zyd(x),d2(x)+zyx,d(x)=0. (3.10)

Combining (3.9) with (3.10), we get [d(x),z]yd2(x)=0 for allxV and y,zU. In particular, [d(x),x]yd2(x)=0. So we get [d(x),x]Ud2(x)=0. The rest of the proof is the same with the end ofTheorem 3.1, and we get the required result.

Now, using the similar techniques, we also prove the following two theorems.

Theorem 3.4. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andFa generalized derivation associated withd=0. Ifd(x)◦F(y) +x◦y=0 for

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Theorem 3.5. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andFa generalized derivation associated withd=0. If either [d(x),F(y)]=[x,y]

or [d(x),F(y)] + [x,y]=0 for allx,y∈U, thenU⊆Z(R).

Theorem 3.6. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andFa generalized derivation associated withd=0. Ifd(x)◦F(y)−xy∈Z(R) for

allx,y∈U, thenU⊆Z(R).

Proof. Assume thatUZ(R), then VZ(R) byLemma 2.4. By hypotheses, we have d(x)◦F(y)−xy∈Z(R). IfF=0, thenxy∈Z(R) for allx,y∈U. In particular, [xy,x]=

0 and hencex[y,x]=0. Replacingyby 2yzwe getxy[z,x]=0 (i.e.,xU[z,x]=0). Hence, byLemma 2.5, eitherx=0 or [z,x]=0. Butx=0 also implies that [z,x]=0; hence for anyx,z∈U, we have [z,x]=0. ThenLemma 2.9gives us the required result. Now we assume thatF=0. For anyx,z∈U, we haved(x)◦F(y)−xy∈Z(R). Replacing yby 2yz and using charR=2, we getd(x)◦F(yz)−xyz∈Z(R) for anyx,y,z∈U. That is, (d(x)F(y)−xy)z+d(x)yd(z)∈Z(R). This implies that [d(x)yd(z),z]=0. Then we haved(x)[yd(z),z] + [d(x),z]yd(z)=0 for anyx,y,z∈U. For anyx∈V, replace yby 2d(x)yin the above relation to get [d(x),z]d(x)yd(z)=0 (i.e., [d(x),z]d(x)Ud(z)=0). Then we have [d(x),z]d(x)=0 ord(z)=0 byLemma 2.5. Now, letU1= {z∈U|[d(x), z]d(x)=0}andU2= {z∈U|d(z)=0}. ThenU1,U2are both additive subgroups ofU andU1∪U2=U. Thus, eitherU=U1orU=U2byLemma 2.6. IfU=U1, replacezby 2zyto get [d(x),z]yd(x)=0, and hence [d(x),z]=0 (especially, [d(x),x]=0) ord(x)=

0. For allx∈Uwe have [d(x),x]=0, thusU⊆Z(R) byLemma 2.7, a contradiction. On the other hand, ifU=U2 thend(U)=0 and henceU⊆Z(R) byLemma 2.10, again a

contradiction.

The following is proved as inTheorem 3.6with necessary variations.

Theorem 3.7. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andFa generalized derivation associated withd=0. Ifd(x)◦F(y) +xy∈Z(R) for

allx,y∈U, thenU⊆Z(R).

Theorem 3.8. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andFa generalized derivation associated withd=0. If [d(x),F(y)]=[x,y] for all x,y∈U, thenU⊆Z(R).

Proof. IfF=0 then [x,y]=0. That is, [U,U]=0, and henceU⊆Z(R) byLemma 2.9. Now we assume thatF=0. Then we have

d(x),F(y)=[x,y] (3.11)

for allx,y∈U. Replacingyby 2yzin (3.11) and using (3.11), we have

F(y)d(x),z+yd(x),d(z)+d(x),yd(z)=y[x,z] (3.12)

for allx,y,z∈U. Now, for anyx∈V, replacezby 2zd(x) in (3.12) and use (3.12) to get

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Again, replaceyby 2tyin (3.13) to get

tyzd(x),d2(x)+tyd(x),zd2(x) +td(x),yzd2(x) +d(x),tyzd2(x)=tyzx,d(x). (3.14)

Combining (3.13) with (3.14), we have[d(x),t]yzd2(x)=0. In particular,[d(x),x]yzd2(x)= 0 for allx∈Vandy,z∈U. Notice that the arguments in the end of the proof ofTheorem 3.1are still valid in the present situation, and hence we get the required results.

We also prove the following as inTheorem 3.8with necessary variations. Theorem 3.9. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andFa generalized derivation associated withd=0. If [d(x),F(y)] + [x,y]=0 for

allx,y∈U, thenU⊆Z(R).

Finally, using the similar techniques as in the above theorems, we prove the following. Theorem 3.10. LetRbe a prime ring with charR=2,Ua Lie ideal such thatu2Ufor all u∈U, andFa generalized derivation associated withd=0. If either [d(x),F(y)]=x◦y

or [d(x),F(y)] +x◦y=0 for allx,y∈U, thenU⊆Z(R).

References

[1] B. Hvala, “Generalized derivations in rings,” Communications in Algebra, vol. 26, no. 4, pp. 1147– 1166, 1998.

[2] M. Ashraf, A. Ali, and R. Rani, “On generalized derivations of prime rings,” Southeast Asian Bulletin of Mathematics, vol. 29, no. 4, pp. 669–675, 2005.

[3] J. Bergen, I. N. Herstein, and J. W. Kerr, “Lie ideals and derivations of prime rings,” Journal of Algebra, vol. 71, no. 1, pp. 259–267, 1981.

[4] P. H. Lee and T. K. Lee, “Lie ideals of prime rings with derivations,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 11, no. 1, pp. 75–80, 1983.

[5] I. N. Herstein, “On the Lie structure of an associative ring,” Journal of Algebra, vol. 14, no. 4, pp. 561–571, 1970.

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