Southeast Asian Bulletin of Mathematics (2001) 25: 379±382
Southeast Asian Bulletin of Mathematics : Springer-Verlag 2001
On Jordan Left Derivations of Lie Ideals in Prime Rings Mohammad Ashraf Department of Mathematics, Faculty of science, King Abdul Aziz University, P.O. Box. 80203, Jeddah 21589, Saudi-Arabia E-mail:
[email protected]
Nadeem-ur-Rehman and Shakir Ali Department of Mathematics, Aligarh Muslim University, Aligarh 202002(India) E-mail:
[email protected]
1991 Mathematics Subject Classi®cation: 16W25, 16N60 Abstract. Let R be a prime ring with characteristic di¨erent from two and U be a Lie ideal of R such that u 2 A U for all u A U. In the present paper it is shown that if d is an additive mappings of R into itself satisfying d
u 2 2ud
u, for all u A U, then either U J Z
R or d
U
0. Keywords: Lie ideals, prime rings, Jordan left derivations, left derivations, torsion free rings.
1. Introduction Throughout the present paper R will denote an associative ring with centre Z
R. Recall that R is 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 A R, then a 0. As usual x; y will denote the commutator xy ÿ yx. An additive subgroup U of R is said to be a Lie ideal of R if u; r A U for all u A U, r A R. An additive mapping d : R ! R is called a derivation (resp. Jordan derivation) if d
xy d
xy xd
y, (resp. d
x 2 d
xx xd
x), holds for all x; y A R. An additive mapping d : R ! R is called a left derivation (resp. Jordan left derivation) if d
xy xd
y yd
x (resp. d
x 2 2xd
x) holds for all x; y A R. One can easily prove that in a noncommutative prime ring any left derivation is zero. In [3], Bresar and Vukman have prove that the existence of a nonzero Jordan left derivation on a prime ring R of char R 0 2; 3 forces R to be commutative. It should be mentioned that the result [3] concerning Jordan left derivation has been improved by Deng [4]. More related results had been obtained in [1], [6] and [8]. In the present paper, our objective is to prove that the following theorem which is a generalization of the result noting about a Jordan left derivations which estab-
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lish that under appropriate restriction on a Lie ideal U of a 2-torsion free prime ring R. 2. Main Theorem Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u 2 A U, for all u A U. If d : R ! R is an additive mapping such that d
u 2 2ud
u for all u A U, then either U J Z
R or d
U
0. Before proceeding the proof of the main theorem we ®rst state a few known results which will be used in subsequent discussion. Lemma 1 ([1, Lemma 2.2]). Let R be a 2-torsion free ring and let U be a Lie ideal of R such that u 2 A U for all u A U. If d : R ! R is an additive mapping satisfying d
u 2 2ud
u for all u A U, then (i) d
uv vu 2ud
v 2vd
u, for all u; v A U. (ii) d
uvu u 2 d
v 3uvd
u ÿ vud
u, for all u; v A U. (iii) d
uvw wvu
uw wud
v 3uvd
w 3wvd
u ÿ vud
w ÿ vwd
u, for all u; v; w A U. (iv) u; vud
u uu; vd
u, forall u; v A U. ÿ (v) u; v d
uv ÿ ud
v ÿ vd
u 0, for all u; v A U. Lemma 2 ([1, Lemma 2.3]). Let R be a 2-torsion free ring and let U be a Lie ideal of R such that u 2 A U, for all u A U. If d : R ! R is an additive mapping satisfying d
u 2 2ud
u for all u A U, then u; vd
u; v 0;
for all u; v A U:
Lemma 3 ([2, Lemma 4]). If U P Z is a Lie ideal of a 2-torsion free prime ring R and a; b A R such that aUb 0, then a 0 or b 0. We begin with the following lemma. Lemma 4. Let R be a 2-torsion free ring and let U be a Lie ideal of R such that u 2 A U, for all u A U. If R admits a Jordan left derivation d : R ! R, then (i) d
u 2 v u 2 d
v
uv vud
u ud
u; v, for all u; v A U. (ii) d
vu 2 u 2 d
v
3vu ÿ uvd
u ÿ ud
u; v, for all u; v A U. Proof. (i) Since uv vu
u v 2 ÿ u 2 ÿ v 2 , we ®nd that uv vu A U, for all u; v A U and uv ÿ vu A U and hence we get 2vu A U. Now, replace v by 2vu in Lemma 1(i), and use the fact that char R 0 2, to get ÿ d
uvu vu 2 2 ud
uv vud
u ; Again, replacing v by 2uv in Lemma 1(i), we get
for all u; v A U:
1
On Jordan Left Derivations of Lie Ideals in Prime Rings
ÿ d
u 2 v uvu 2 ud
uv uvd
u ;
381
for all u; v A U:
2
Now, subtracting equation (1) from equation (2) yields that ÿ d
u 2 v ÿ vu 2 2 ud
u; v u; vd
u ;
for all u; v A U:
3
Replacing u by u 2 in Lemma 1(i), we have ÿ d
u 2 v vu 2 2 u 2 d
v 2vud
u ;
for all u; v A U:
4
Hence, adding (3) and (4) and using the fact that char R 0 2, we obtain d
u 2 v u 2 d
v
uv vud
u ud
u; v;
for all u; v A U:
(ii) As in the proof of the case (i) subtracting (3) from (4), we ®nd that d
vu 2 u 2 d
v
3vu ÿ uvd
u ÿ ud
u; v;
for all u; v A U:
Proof of the Main Theorem. Suppose that on contrary U P Z
R. By Lemma 1(iv), we have
u 2 v ÿ 2uvu vu 2 d
u 0;
for all u; v A U:
5
Replacing u by u; w in (5), we get u; w 2 vd
u; w ÿ 2u; wvu; wd
u; w vu; w 2 d
u; w 0;
for all u; v; w A U:
Now, application of Lemma 2 yields that u; w 2 Ud
u; w
0. Hence by Lemma 3, either u; w 2 0 or d
u; w 0. If u; w 2 0, for all u; w A U, then linearizing the above relation on w, we get u; wu; v u; vu; w 0;
for all u; v; w A U:
6
For any u; v A U, uv vu
u v 2 ÿ u 2 ÿ v 2 A U and uv ÿ vu A U and hence 2vu A U. Replacing v by 2vu in (6) and since char R 0 2, we ®nd that u; wu; vu u; vuu; w 0;
for all u; v; w A U;
and hence by application of (6), we obtain u; vu; u; w 0. Now replace v by 2vv1 , to get u; vv1 u; u; w 0, for all u; v; v1 ; w A U i.e. u; vUu; u; w
0. Thus by Lemma 3, we ®nd that for each u A U, either u; v 0 or u; u; w 0. If u; u; w 0, for all w A U, then replacing w by 2vw, we get u; vu; w 0, for all v; w A U. Again replace w by 2wv, to get u; vwu; v 0, for all v; w A U i.e. u; vUu; v
0, for all v A U and hence by Lemma 3, we get u; v 0. Thus, in
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both the cases we ®nd that u; v 0. Therefore, U is a commutative Lie ideal of R, and hence by using the same arguments as used in the proof of Lemma 1.3 of [5] U J Z
R, a contradiction. Hence, we consider the case d
u; w 0, for all u; w A U. So we have ÿ ÿ 2d
wuu d
wuu u
wu 2fu 2 d
w uwd
u wud
ug: Since R is a 2-torsion free, we obtain ÿ d
wuu u 2 d
w uwd
u wud
u for all u; w A U:
7
Using Lemma 4(ii) and (7), we get u; wd
u 0;
for all u; w A U:
8
Now, replacing w by 2wv in (8) and since char R 0 2, we get u; wvd
u 0 i.e. u; wUd
u
0. Thus, by Lemma 3, we ®nd that for each u A U either u; w 0 or d
u 0. If u; w 0, then by using the similar arguments as above we get U J Z
R, again a contradiction. Hence the remaining possibility that d
u 0 ie. d
U
0. This completes the proof of the theorem. References 1. Ashraf, M., Rehman, N.: On Lie ideals and Jordan left derivations of prime rings, Arch. Math. (Brno) 36, 201±206 (2000). 2. Bergen, J., Herstein, I.N., Kerr, J.W.: Lie ideals and derivations of prime rings, J. Algebra 71, 259±267 (1981). 3. Bresar, M., Vukman, J.: On left derivations and related mappings, Proc. Amer. Math. Soc. 110, 7±16 (1990). 4. Deng, Q.: On Jordan left derivations, Math. J. Okayama Univ. 34, 145±147 (1992). 5. Herstein, I.N.: Topics in ring theory, Univ. of Chicago Press, Chicago, 1969. 6. Jun, K.W., Kim, B.D.: A note on Jordan left derivations, Bull. Korean Math. Soc. 33(2), 221±228 (1996). 7. Posner, E.C.: Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093±1100 (1957). 8. Vukman, J.: Jordan left derivations on semiprime rings, Math. J. Okayama Univ. 39, 1± 6 (1997).