PII. S0161171204309075 http://ijmms.hindawi.com © Hindawi Publishing Corp.
ON JORDAN IDEALS AND LEFT
(θ, θ)
-DERIVATIONS
IN PRIME RINGS
S. M. A. ZAIDI, MOHAMMAD ASHRAF, and SHAKIR ALI
Received 8 September 2003
LetRbe a ring andSa nonempty subset ofR. Suppose thatθandφare endomorphisms of R. An additive mappingδ:R→Ris called a left(θ, φ)-derivation (resp., Jordan left(θ, φ) -derivation) onSifδ(xy)=θ(x)δ(y)+φ(y)δ(x)(resp.,δ(x2)=θ(x)δ(x)+φ(x)δ(x)) holds for allx, y∈S. Suppose thatJ is a Jordan ideal and a subring of a 2-torsion-free prime ringR. In the present paper, it is shown that ifθis an automorphism of Rsuch thatδ(x2)=2θ(x)δ(x)holds for allx∈J, then eitherJ⊆Z(R)orδ(J)=(0). Further, a study of left(θ, θ)-derivations of a prime ringRhas been made which acts either as a homomorphism or as an antihomomorphism of the ringR.
2000 Mathematics Subject Classification: 16W25, 16N60, 16U80.
1. Introduction. Throughout the present paper,R will denote an associative ring with centreZ(R). We will write for allx, y∈R,[x, y]=xy−yxandx◦y=xy+yx for the Lie product and Jordan product, respectively. A ring R is said to be prime if aRb =(0) implies that a =0 or b =0. A ring R is said to be 2-torsion-free if whenever 2a=0, with a∈ R, then a= 0. An additive subgroup J of R is said to be a Jordan ideal of R if u◦r ∈J, for all u∈J and r ∈R. An additive mapping d:R→Ris called a derivation (resp., Jordan derivation) if d(xy)=d(x)y+xd(y) (resp.,d(x2)=d(x)x+xd(x)) holds for allx, y∈R. Letθ,φbe endomorphisms of
R. An additive mappingf :R→R is called a (θ, φ)-derivation (resp., Jordan (θ, φ) -derivation) iff (xy)=f (x)θ(y)+φ(x)f (y)(resp.,f (x2)=f (x)θ(x)+φ(x)f (x)) holds, for allx, y∈R. Of course a(1,1)-derivation (resp., a Jordan(1,1)-derivation) is a derivation (resp., a Jordan derivation) onR, where 1 is the identity mapping on R. We will make use of the following basic commutator identities without any specific mention:
[xy, z]=x[y, z]+[x, z]y, [x, yz]=y[x, z]+[x, y]z. (1.1)
An additive mappingδ:R→R is called a left derivation (resp., Jordan left deriva-tion) if δ(xy) =xδ(y)+yδ(x) (resp., δ(x2)=2xδ(x)) holds for all x, y ∈R. In view of the definition of a (θ, φ)-derivation, the notion of left(θ, φ)-derivation can be extended as follows: let θ, φ be endomorphisms ofR and let S be a nonempty subset ofR. An additive mapping δ:R→R is called a left (θ, φ)-derivation (resp., Jordan left(θ, φ)-derivation) onS if δ(xy)=θ(x)δ(y)+φ(y)δ(x)(resp., δ(x2)=
where 1 is the identity mapping onR. In [5], Brešar and Vukman have proved that the existence of a nonzero Jordan left derivation on a prime ringRof charR≠2,3 forces Rto be commutative. It should be mentioned that the result obtained in [5] concerning Jordan left derivation has been improved by Deng [7]. Some more related results can be seen in [1,3,5,7,9]. It is easy to see that every left derivation on a ringRis a Jordan left derivation. However, in general, a Jordan left derivation need not be a left derivation. The following example justifies this statement.
Example1.1. LetRbe a commutative ring and leta∈Rsuch thatxax=0 for all
x∈Rbutxay≠0, for somexandy,x≠y. Define a mapδ:R→Ras follows:
δ(x)=xa+ax. (1.2)
Thenδis a Jordan left derivation but not a left derivation.
In the present paper, first it is shown that every Jordan left(θ, θ)-derivation on a Jordan idealJof a 2-torsion-free prime ring is a left(θ, θ)-derivation onJ. Finally, we will study the behaviour of left(θ, θ)-derivation on a prime ring which also acts either as a homomorphism or an antihomomorphism of the underlying ring.
2. Preliminary results. We begin with the following lemmas which are essential in developing the proof of our main result.
Lemma2.1[6, Lemma 4]. LetGandHbe additive groups and letRbe a2-torsion-free
ring. Letf:G×G→Handg:G×G→Rbe biadditive mappings. Suppose that for each paira, b∈Geitherf (a, b)=0org(a, b)2=0. In this case, eitherf=0org(a, b)2=0 for alla, b∈G.
If J is assumed to be a Jordan ideal and a subring of a ringR, then using similar techniques as used in the proofs of Lemmas 2.2 and 2.3 of [1], one can easily obtain the following lemma.
Lemma 2.2. Let R be a 2-torsion-free ring, let J be a Jordan ideal and a subring
of R. If θ is an endomorphism ofR andδ:R→R is an additive mapping satisfying δ(u2)=2θ(u)δ(u), for allu∈J, then
(i) δ(uv+vu)=2θ(u)δ(v)+2θ(v)δ(u), for allu, v∈J,
(ii) δ(uvu)=θ(u2)δ(v)+3θ(u)θ(v)δ(u)−θ(v)θ(u)δ(u), for allu, v∈J, (iii) δ(uvw + wvu) = (θ(u)θ(w) + θ(w)θ(u))δ(v) + 3θ(u)θ(v)δ(w) +
3θ(w)θ(v)δ(u)−θ(v)θ(u)δ(w)−θ(v)θ(w)δ(u), for allu, v, w∈J, (iv) [θ(u), θ(v)]θ(u)δ(u)=θ(u)[θ(u), θ(v)]δ(u), for allu, v∈J,
(v) [θ(u), θ(v)](δ(uv)−θ(u)δ(v)−θ(v)δ(u))=0, for allu, v∈J.
Lemma2.3. LetRbe a2-torsion-free ring,Ja Jordan ideal and a subring ofR. Ifθis an
endomorphism ofRandδ:R→Ris an additive mapping satisfyingδ(u2)=2θ(u)δ(u) for allu∈J,then
(i) [θ(u), θ(v)]δ([u, v])=0, for allu, v∈J;
We begin with the following lemma.
Lemma2.4. IfRis a ring andJa nonzero Jordan ideal ofR, then2[R, R]J⊆Jand
2J[R, R]⊆J.
Proof. Letx, y∈R andu∈J. Thenu◦[x, y]−(u◦x)◦y+(u◦y)◦x∈J. This
implies that uxy−uyx+xyu−yxu−uxy−xuy−yux−yxu+uyx+yux+ xuy+xyu∈Jand hence 2[x, y]u∈J, for allx, y∈R, andu∈J, that is, 2[R, R]J⊆J. Similarly, it is easy to see that 2u[x, y]=(u◦y)◦x−u◦[x, y]−(u◦x)◦y∈J, for allx, y∈Randu∈J, and hence 2J[R, R]⊆J.
Lemma2.5. LetR be a prime ring andJa nonzero Jordan ideal ofR. Ifa∈Rand
aJ=(0) (orJa=(0)), thena=0.
Proof. SinceJis a Jordan ideal ofR,u◦x∈J, for allx∈Randu∈J. By
hypothe-ses, we havea(u◦x)=0, for allx∈R,u∈J, and hence we getaxu=0, for allx∈R, u∈J, that is,aRJ=(0). SinceJis a nonzero Jordan ideal andRis prime, the above relation yields thata=0.
IfJa=(0), then using similar arguments with necessary variations, we get the re-quired result.
Lemma2.6. LetRbe a2-torsion-free prime ring andJa nonzero Jordan ideal ofR.
IfaJb=(0), thena=0orb=0.
Proof. ByLemma 2.4, we find that 2[R, R]J⊆J. Thus, for anyx, y∈Randu∈J,
we have 2a[x, y]ub=0. This implies that
a[x, y]ub=0, ∀x, y∈R, u∈J. (2.1)
Replacingy byyain the above expression, we geta[x, ya]ub=0, for allx, y∈R and u∈J oray[x, a]ub+a[x, y]aub=0. Now, using the fact thataJb=(0), we find that ay[x, a]ub=0, for all x, y ∈R and u∈J and hence aR[x, a]ub=(0). Thus, primeness ofRforces that eithera=0 or[x, a]ub=0. If[x, a]ub=0, for all x∈R,u∈J, then by our hypotheses we haveaxub=0, for allx∈R,u∈J, that is, aRub=(0). Again, primeness ofRgives that eithera=0 orub=0. Ifub=0, for all u∈J, then byLemma 2.5, we getb=0.
Lemma2.7. LetRbe a2-torsion-free prime ring andJa nonzero Jordan ideal ofR.
IfJis a commutative Jordan ideal, thenJ⊆Z(R).
Proof. ByLemma 2.4, we have 2[R, R]J⊆J. Thus, for anyx, y∈Randu, v∈J, we
The next lemma can be regarded as a generalization of a lemma due to Smiley [8] for Jordan ideals of a prime ring.
Lemma2.8. LetR be a2-torsion-free prime ring and letJ be a Jordan ideal and a
subring ofR such that[u, v]2=0, for allu, v∈J. ThenJ is commutative and hence central.
Proof. By hypothesis, we have[u, v]2=0, for allu, v∈J. On linearizing, we get
[u, v][u, w]+[u, w][u, v]=0, for allu, v, w∈J. Replacingvbyvuin the above ex-pression and using it, we obtain[u, v][u, [u, w]]=0, for allu, v, w∈J. Again, replac-ingvbyvv1in latter relation, we find that[u, v]v1[u, [u, w]]=0, that is,[u, v]J[u, [u,
w]]= (0), for all u, v, w ∈J. Thus by Lemma 2.6, we have for each u ∈J either [u, v]=0 or[u, [u, w]]=0, for allu, v, w∈J. If[u, [u, w]]=0, for allu, w∈J, then on replacingwbywv, we get[u, w][u, v]=0, for allu, v, w∈J. Again, replacingv byvw, we have[u, w]v[u, w]=0, for allu, v, w∈Jand hence[u, w]J[u, w]=(0), for allw∈J. Again, byLemma 2.6, we obtain[u, w]=0. Thus in both cases we find that[u, w]=0, for allu, w∈J. Thus,Jis commutative, and byLemma 2.7,Jis central, that is,J⊆Z(R).
Lemma2.9. LetR be a2-torsion-free ring, J a Jordan ideal and a subring ofR. If
δ:R→Ris an additive mapping satisfyingδ(u2)=2θ(u)δ(u), for allu∈U, then (i) δ(u2v) = θ(u2)δ(v)+(θ(u)θ(v)+θ(v)θ(u))δ(u)+θ(u)δ([u, v]), for all
u, v∈J,
(ii) δ(vu2)= θ(u2)δ(v)+(3θ(v)θ(u)−θ(u)θ(v))δ(u)−θ(u)δ([u, v]), for all
u, v∈J.
Proof. (i) ReplacingvbyvuanduvinLemma 2.2(i), we find that
δuvu+vu2=2θ(u)δ(vu)+θ(v)θ(u)δ(u), ∀u, v∈J, (2.2)
δu2v+uvu=2θ(u)δ(uv)+θ(u)θ(v)δ(u), ∀u, v∈J. (2.3)
Now, subtracting (2.2) from (2.3), we get
δu2v−vu2=2θ(u)δ[u, v]+θ(u), θ(v)δ(u), ∀u, v∈J. (2.4)
Replacingubyu2inLemma 2.2(i), we have
δu2v+vu2=2θu2δ(v)+2θ(v)uδu2
=2θu2δ(v)+4θ(v)θ(u)δ(u), ∀u, v∈J. (2.5)
Hence adding (2.4), (2.5) and using the fact that charR≠2, we obtain
δu2v=θu2δ(v)+θ(u)θ(v)+θ(v)θ(u)δ(u)
(ii) As in the proof of the case (i), subtracting (2.4) from (2.5), we find that
δvu2=θu2δ(v)+3θ(v)θ(u)−θ(u)θ(v)δ(u)
−θ(u)δ[u, v], ∀u, v∈J. (2.7)
3. Left derivation on Jordan ideal of a prime ring. In [3], there is a more general result which implies that in a 2-torsion-free prime ringR, the existence of a nonzero Jordan left derivation on a Lie idealUofRforces that eitherU⊆Z(R)orδ(U )=(0). In the present section, we attempt to generalize the above-mentioned result for Jordan left(θ, θ)-derivation which acts on a Jordan ideal of the ring.
Theorem3.1. LetRbe a2-torsion-free prime ring and letJbe a Jordan ideal and
a subring of R. If θ is an automorphism of R andδ:R→R is an additive mapping satisfyingδ(u2)=2θ(u)δ(u), for allu∈J, then eitherJ⊆Z(R)orδ(J)=(0).
Proof. Suppose thatJ⊆Z(R). ByLemma 2.2(iv), we have
θ(u), θ(v)θ(u)δ(u)=θ(u)θ(u), θ(v)δ(u), ∀u, v∈J. (3.1)
This implies that
θu2θ(v)−2θ(u)θ(v)θ(u)+θ(v)θu2δ(u)=0, ∀u, v∈J. (3.2)
Replacinguby[u, w]in (3.2), we get
θ[u, w]2θ(v)δ[u, w]−2θ[u, w]θ(v)θ[u, w]δ[u, w]
+θ(v)θ[u, w]2δ[u, w]=0, (3.3)
for allu, v, w∈J. Now, application ofLemma 2.3(i) yields thatθ([u, w]2)θ(v)δ([u, w])
=(0), for allu, v, w∈J. Sinceθis an automorphism ofR, the latter expression gives [u, w]2Jθ−1(δ([u, w]))=(0). Hence, byLemma 2.6, we find that for each pairu, w∈J, either [u, w]2= 0 or θ−1(δ([u, w]))= 0. Note that the mappings (u, w)[u, w] and (u, w) θ−1(δ([u, w])) satisfy the requirements of Lemma 2.1. Hence, either
[u, w]2=0, for all u, w∈J, or θ−1(δ([u, w]))=0, for all u, w∈J. If[u, w]2=0, for allu, w∈J, then by application ofLemma 2.8,Jis commutative and hence central, that is,J⊆Z(R), a contradiction. Now, we consider the caseθ−1(δ([u, w]))=0, then
δ([u, w])=0, that is,δ(uw)=δ(wu), for allu, w∈J. In view of Lemma 2.2(i), we have
2δ(wu)u=δ(wu)u+u(wu)
=2θ(w)θ(u)δ(u)+2θ(u)δ(wu+uw)
=2θu2δ(w)+θ(u)θ(w)δ(u)+θ(w)θ(u)δ(u), ∀u, w∈J. (3.4)
[u, w]=0 orθ−1(δ(u))=0. Sinceθ is an automorphism, we have either[u, w]=0 or δ(u) =0, for all w ∈ J. Now let J1= {u∈ J | [u, w]= 0, for allw ∈ J} and
J2= {u∈J|δ(u)=0}. Clearly,J1 andJ2 are additive subgroups ofJ whose union isJ. Hence, by Brauer’s trick, eitherJ=J1orJ=J2. IfJ=J1, then[u, w]=0, for all
u, w∈J, that is,Jis commutative, and hence byLemma 2.7,J⊆Z(R), again a contra-diction. Hence, we have the remaining possibility thatδ(u)=0, for allu∈J, that is, δ(J)=(0). This completes the proof of the theorem.
Remark3.2. In the hypotheses of the above theorem, if we assume only thatJis a
subring ofR, then neitherJis central norδ(J)=(0). This is shown by the following example.
Example3.3. LetSbe a ring such that the square of each element inSis zero, but the
product of some elements inSis nonzero. Further, suppose thatR= {x y0 0
|x, y∈S}. ConsiderJ= {0y
0 0
|y∈S}, thenJis a subring ofR. Define mappingsδ:R→R and θ:R→Ras follows:
δ x y 0 0 = 0 y 0 0 , θ x y 0 0 =
x −y
0 0
. (3.5)
It is easy to verify thatδ is a Jordan left(θ, θ)-derivation, but neitherJ⊆Z(R)nor δ(J)=(0).
Corollary3.4. LetRbe a2-torsion-free prime ring. Ifδ:R→Ris a nonzero additive
mapping satisfyingδ(x2)=2xδ(x), for allx∈R,thenRis commutative.
The following example demonstrates that to haveRprime is essential in the hypoth-esis of the above result.
Example3.5. Consider a ringR, as inExample 3.3, and define mappingsδ:R→R
andθ:R→Ras follows:
δ x y 0 0 = 0 −x
0 0 , θ x y 0 0 =
x −y
0 0
. (3.6)
Then, withJ=R, it can be easily seen thatδ(x2)=2θ(x)δ(x), for allx∈R, butR is not commutative.
4. Left derivation as a homomorphism or as an antihomomorphism. LetS be a nonempty subset of a ring R and da derivation of R. Ifd(xy)=d(x)d(y) (resp., d(xy)=d(y)d(x)) holds for allx, y∈S, then we say thatdacts as a homomorphism (resp., antihomomorphism) onS.
Theorem4.1. LetR be a prime ring,I a nonzero right ideal ofR, and letθ,φ be
automorphisms ofR. Suppose thatδ:R→Ris a(θ, φ)-derivation ofR. (i) Ifδacts as a homomorphism onI, thenδ=0onR.
(ii) Ifδacts as an antihomomorphism onI, thenδ=0onR.
In the present section, our objective is to extend the above result for left(θ, θ) -deriva-tion of a prime ringRwhich acts as a homomorphism or as an antihomomorphism on a Jordan idealJofR. In fact, we prove the following theorem.
Theorem 4.2. Let R be a2-torsion-free prime ring andJ a nonzero Jordan ideal
and a subring of R. Suppose thatθ is an automorphism of R andδ:R →R is a left (θ, θ)-derivation ofR.
(i) Ifδacts as a homomorphism onJ, thenδ=0onR. (ii) Ifδacts as an antihomomorphism onJ, thenδ=0onR.
Proof. (i) By our hypotheses, we have
δ(u)δ(v)=δ(uv)=θ(u)δ(v)+θ(v)δ(u), ∀u, v∈J. (4.1)
Replacingubyuvin (4.1), we find that
δ(uv)δ(v)=θ(uv)δ(v)+θ(v)δ(uv), ∀u, v∈J. (4.2)
Now, application of (4.1) yields thatθ(u)δ(v)δ(v)=θ(uv)δ(v), for allu, v∈J. This implies that
θ(u)δ(v)−θ(v)δ(v)=0, ∀u, v∈J. (4.3)
Thus,θ(J)(δ(v)−θ(v))δ(v)=(0), for allv∈J. Sinceθis an automorphism andJis a nonzero Jordan ideal ofR,θ(J)is also a nonzero Jordan ideal ofR. Application of Lemma 2.6yields that(δ(v)−θ(v))δ(v)=0, for allv∈Jand henceδ(v2)=θ(v)δ(v), for allv∈J. Sinceδis a left(θ, θ)-derivation, we haveθ(v)δ(v)=0, for allv∈J. On linearizing the latter relation, we find that
θ(v)δ(u)+θ(u)δ(v)=0, ∀u, v∈J. (4.4)
Again, replacingu byvuin (4.4), we getθ(v)θ(u)δ(v)=0, for allu, v∈J, that is, vJθ−1(δ(v))=(0), for allv∈J. Application ofLemma 2.6yields that eitherv=0 or
θ−1(δ(v))=0. Butv=0 also gives thatθ−1(δ(v))=0, that is,δ(v)=0, for allv∈J. Further, replacev byv◦r to get 2θ(v)δ(r )=0,for allv∈J andr ∈R. SinceR is 2-torsion-free andθ(J)is a nonzero Jordan ideal ofR,application ofLemma 2.6yields the required result.
(ii) Ifdacts as an antihomomorphism onJ, then
δ(u)δ(v)=δ(vu)=θ(v)δ(u)+θ(u)δ(v)
=θ(u)δ(v)+θ(v)δ(u)=δ(uv)=δ(v)δ(u), (4.5)
Remark 4.3. We feel that Theorem 3.1 (resp., Theorem 4.2) could be proved for
Jordan left(θ, φ)-derivation (resp., left(θ, φ)-derivation) of a prime ring. However, we did not succeed to settle it.
Acknowledgments. The authors are greatly indebted to the referees for their
valuable suggestions. This research is supported by the University Grants Commission (UGC), India Grant F-510/3/DSA/98(SAP-I).
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S. M. A. Zaidi: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail address:[email protected]
Mohammad Ashraf: Department of Mathematics, Faculty of Science, King Abdul Aziz Univer-sity, P.O. Box 80203, Jeddah 21589, Saudi Arabia
E-mail address:[email protected]
