# Remarks on Generalized Derivations in Prime and Semiprime Rings

Authors: Basudeb Dhara
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

LetR be a ring with center Z and I a nonzero ideal of R. An additive mapping F:R→R is called a generalized derivation of R if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) for all x,y∈R. In the present paper, we prove that if F([x,y])=±[x,y] for all x,y∈I or F(x∘y)=±(x∘y) for all x,y∈I, then the semiprime ring R must contains a nonzero central ideal, provided d(I)≠0. In case R is prime ring, R must be commutative, provided d≠0. The cases (i) F([x,y])±[x,y]∈Z and (ii) F(x∘y)±(x∘y)∈Z for all x,y∈I are also studied.


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## Body

## 1. Introduction

LetR 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 xy+yx. We will make extensive use of basic commutator identities [xy,z]=[x,z]y+x[y,z], [x,yz]=[x,y]z+y[x,z]. An additive mapping d from R to R is called a derivation of R if d(xy)=d(x)y+xd(y) 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 g(xy)=g(x)y+xd(y) 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 [F(x),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. In [1], Daif and Bell proved that if R is a semiprime ring with a nonzero ideal K and d is a derivation of R such that d([x,y])=±[x,y] for all x,y∈K, then K is central ideal. In particular, if K=R, then R is commutative. Recently, Quadri et al. [2] generalized this result replacing derivation d with a generalized derivation in a prime ring R. More precisely, they proved the following.LetRbe a prime ring andIa nonzero ideal ofR. IfRadmits a generalized derivationFassociated with a nonzero derivationdsuch that any one of the following holds:  (i)
F([x,y])=[x,y]for allx,y∈I,  (ii)  F([x,y])=-[x,y]for allx,y∈I,  (iii)  F(x∘y)=(x∘y)for allx,y∈I; (iv)F(x∘y)=-(x∘y)for 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 (see [3, Lemma 1.1.5]or [1, Lemma 2]).
(a) IfR is a semiprime ring, the center of a nonzero one-sided ideal is contained in the center of R, in particular, any commutative one-sided ideal is contained in the center of R.
(b) IfR is a prime ring with a nonzero central ideal, then R must be commutative.Lemma 2.2 (see [1, Lemma 1]).
LetR be a semiprime ring and I a nonzero ideal of R. If z∈R and z centralizes [I,I], then z centralizes I.Lemma 2.3 (see [4, Theorem 3]).
LetR be a semiprime ring and U a nonzero left ideal of R. If R admits a derivation d which is nonzero on U and centralizing on U, then R contains a nonzero central ideal.Now we begin with the theorem.Theorem 2.4.
LetR be a semiprime ring, I a nonzero ideal of R and F a generalized derivation of R associated with a derivation d of R such that d(I)≠0. If F([x,y])=±[x,y] for all x,y∈I, then R contains a nonzero central ideal.Proof.
By our assumption, we have that(2.1)F([x,y])=±[x,y]
for all x,y∈I. If F(I)=0, then we find that [x,y]=0 for all x,y∈I, that is, I is commutative. Then, by Lemma 2.1, I⊆Z and thus we obtain our conclusion.
Next assume thatF(I)≠0. Putting y=yx in (2.1), we get that(2.2)F([x,y]x)=±[x,y]x.
Since F is a generalized derivation of R associated with a derivation d of R, (2.2) gives
(2.3)F([x,y])x+[x,y]d(x)=±[x,y]x.
Using (2.1), it reduces to
(2.4)[x,y]d(x)=0
for all x,y∈I. Now putting y=d(x)y in (2.4), we get
(2.5)0=[x,d(x)y]d(x)=d(x)[x,y]d(x)+[x,d(x)]yd(x).
Using (2.4), it gives
(2.6)0=[x,d(x)]yd(x)
for all x,y∈I. Now we put y=yx in (2.6) and obtain that
(2.7)0=[x,d(x)]yxd(x)
for all x,y∈I. Right multiplying (2.6) by x and then subtracting from (2.7), we get
(2.8)0=[x,d(x)]y[x,d(x)]
for all x,y∈I. This implies for all x∈I that ([x,d(x)]I)2=0 and so [x,d(x)]I=0, forcing [x,d(x)]∈I∩Ann(I)=0. Then by Lemma 2.3, R contains a nonzero central ideal.Corollary 2.5.
LetR be a prime ring, I a nonzero ideal of R and F a generalized derivation of R. If F([x,y])=±[x,y] for all x,y∈I, then R is commutative or F(x)=±x for all x∈I.Proof.
Letd be the associated derivation of F. By Theorem 2.4, we conclude that either d(I)=0 or R is commutative. Assume that R is not commutative. Then d(I)=0. Since R is a prime ring, d(I)=0 implies d(R)=0 and hence F(xy)=F(x)y for all x,y∈R. Set G(x)=F(x)∓x for all x∈R. Then G(xy)=G(x)y for all x∈R. Now, our assumption F([x,y])=±[x,y] gives F(x)y-F(y)x=±(xy-yx), that is, G(x)y-G(y)x=0 for all x,y∈I. Thus using G(x)y=G(y)x, we have G(x)yz=G(y)xz=G(xz)y=G(x)zy, that is, G(x)[y,z]=0 for all x,y,z∈I. Thus 0=G(I)[I,I]=G(IR)[I,I]=G(I)R[I,I]. Since R is prime, this implies G(I)=0 or I is commutative. By Lemma 2.1, I commutative implies that R is commutative, a contradiction. Thus G(I)=0 which gives G(x)=F(x)∓x=0 for all x∈I.Theorem 2.6.
LetR be a semiprime ring, I a nonzero ideal of R and F a generalized derivation of R associated with a derivation d of R such that d(I)≠0. If F(x∘y)=±(x∘y) for all x,y∈I, then R contains a nonzero central ideal.Proof.
IfF(I)=0, then by our assumption we have that x∘y=0, that is, xy+yx=0 for all x,y∈I. This implies that x(yz)=-(yz)x=-y(zx)=y(xz)=(yx)z=-(xy)z for all x,y,z∈I and so 2I3=0, forcing 2I=0. Therefore, for all x,y∈I, xy+yx=0 gives xy=yx, that is, I is commutative. Then by Lemma 2.1, I⊆Z and thus we obtain our conclusion.Next assume thatF(I)≠0. Then for any x,y∈I, we have  (2.9)F(xy+yx)=±(xy+yx).
Since F is a generalized derivation associated with a derivation d, above expression yields  (2.10)F(x)y+xd(y)+F(y)x+yd(x)=±(xy+yx).
Putting y=yx in (2.10), we have  (2.11)F(x)yx+x(d(y)x+yd(x))+(F(y)x+yd(x))x+yxd(x)=±(xyx+yx2).
Right multiplying (2.10) by x and then subtracting from (2.11), we get  (2.12)xyd(x)+yxd(x)=0
for all x,y∈I. Replacing y with d(x)y in (2.12) and then again using (2.12) we find that  (2.13)[x,d(x)]yd(x)=0.
Again replacing y with yx in (2.13) and then using (2.13) we obtain  (2.14)[x,d(x)]y[x,d(x)]=0
for all x,y∈I, which is the same identity as (2.8) in the proof of Theorem 2.4. Thus by the same argument as in the proof of Theorem 2.4, we conclude that R contains a nonzero central ideal.Corollary 2.7.
LetR be a prime ring, I a nonzero ideal of R and F a generalized derivation of R. If F(x∘y)=±(x∘y) for all x,y∈I, then R is commutative or F(x)=±x for all x∈I.Proof.
Letd be the associated derivation of F. By Theorem 2.6, we conclude that either d(I)=0 or R is commutative. If R is not commutative, then d(I)=0. Since R is a prime ring, d(I)=0 implies d(R)=0 and hence F(xy)=F(x)y for all x,y∈R. Set G(x)=F(x)∓x for all x∈R. Then G(xy)=G(x)y for all x∈R. Now, our assumption F(x∘y)=±(x∘y) gives F(x)y+F(y)x=±(xy+yx), that is, G(x)y+G(y)x=0 for all x,y∈I. Thus using G(x)y=-G(y)x, we have G(x)yz=-G(y)xz=G(xz)y=G(x)zy, that is, G(x)[y,z]=0 for all x,y,z∈I. Thus 0=G(I)[I,I]=G(IR)[I,I]=G(I)R[I,I]. Since R is prime, this implies G(I)=0 or I is commutative. By Lemma 2.1, I commutative implies that R is commutative, a contradiction. Therefore, G(I)=0 and hence G(x)=F(x)∓x=0 for all x∈I.Theorem 2.8.
LetR be a semiprime ring with center Z≠{0}, I a nonzero ideal of R and F a generalized derivation of R associated with a derivation d of R. If F([x,y])±[x,y]∈Z for all x,y∈I, then Id(Z)⊆Z.Proof.
We have(2.15)F([x,y])±[x,y]∈Z
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 in (2.15) and then we get
(2.16)F([x,y]z)±[x,y]z=F([x,y])z+[x,y]d(z)±[x,y]z={F([x,y])±[x,y]}z+[x,y]d(z)∈Z.
By (2.15), we have [x,y]d(z)∈Z for all x,y∈I. Since d(z)∈Z, this gives that for any r∈R, [r,[x,y]d(z)]=0 which implies [rd(z),[x,y]]=0 for all x,y∈I. By Lemma 2.2, [rd(z),x]=0 for all x∈I. Since d(z)∈Z, this gives [r,xd(z)]=0 for all r∈R and for all x∈I. Thus, xd(z)∈Z, that is, Id(Z)⊆Z.Corollary 2.9.
LetR be a prime ring with center Z≠{0}, I a nonzero ideal of R and F a generalized derivation of R associated with a derivation d. If d(Z)≠{0} and F([x,y])±[x,y]∈Z for all x,y∈I, then R is commutative.Proof.
Sinced(Z)⊆Z and Z contains no nonzero elements which are zero divisors, we have from Theorem 2.8 that I⊆Z. Then by Lemma 2.1(b), we obtain our conclusion.Theorem 2.10.
LetR be a semiprime ring with center Z≠{0}, I a nonzero ideal of R and F a generalized derivation of R associated with a derivation d of R. If F(x∘y)±(x∘y)∈Z for all x,y∈I, then Id(Z)⊆Z.Proof.
We have(2.17)F(x∘y)±(x∘y)∈Z
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 in (2.17) and then we get
(2.18)F((x∘y)z)±(x∘y)z=F((x∘y))z+(x∘y)d(z)±(x∘y)z={F(x∘y)±(x∘y)}z+(x∘y)d(z)∈Z.
By (2.17), we have (x∘y)d(z)∈Z that is (xy+yx)d(z)∈Z for all x,y∈I. Now putting y=yr and x=rx, r∈R, respectively, we obtain that (xyr+yrx)d(z)∈Z and (rxy+yrx)d(z)∈Z. Subtracting these two results yields [xyd(z),r]∈Z for all x,y∈I and for all r∈R. This gives
(2.19)[[xyd(z),r],s]=0
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
(2.20)0=[[xyd(z),r],s]=-[[r,s],xyd(z)]-[[s,xyd(z)],r].
By using (2.19), it reduces to
(2.21)[[r,s],xyd(z)]=0
for all r,s∈R and for all x,y∈I. By Lemma 2.2, this implies that [xyd(z),r]=0, that is, [I2d(z),R]=0. Thus [[I,I],Id(z)]=0 and then again by Lemma 2.2, [I,Id(z)]=0. This yields 0=[IR,Id(z)]=I[R,Id(z)] which implies Id(z)⊆Z, since [R,Id(z)]⊆I∩Ann(I)=0. Since z is any nonzero element in Z, we conclude that Id(Z)⊆Z.



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*Source: 646587-2010-12-20.xml*