On Jordan Left Derivations of Lie Ideals in Prime Rings - Springer Link

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Keywords: Lie ideals, prime rings, Jordan left derivations, left derivations, torsion free rings. 1. Introduction. Throughout the present paper R will denote an ...
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 † ˆ 2ud…u†, for all u A U, then either U J Z…R† or d…U† ˆ …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 ˆ …0† implies that a ˆ 0 or b ˆ 0. A ring R is said to be 2-torsion free if whenever 2a ˆ 0, with a A R, then a ˆ 0. As usual ‰x; yŠ will denote the commutator xy ÿ yx. An additive subgroup U of R is said to be a Lie ideal of R if ‰u; rŠ 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† ˆ d…x†y ‡ xd… y†, (resp. d…x 2 † ˆ d…x†x ‡ 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† ˆ xd…y† ‡ yd…x† (resp. d…x 2 † ˆ 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 † ˆ 2ud…u† for all u A U, then either U J Z…R† or d…U† ˆ …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 † ˆ 2ud…u† for all u A U, then (i) d…uv ‡ vu† ˆ 2ud…v† ‡ 2vd…u†, for all u; v A U. (ii) d…uvu† ˆ u 2 d…v† ‡ 3uvd…u† ÿ vud…u†, for all u; v A U. (iii) d…uvw ‡ wvu† ˆ …uw ‡ wu†d…v† ‡ 3uvd…w† ‡ 3wvd…u† ÿ vud…w† ÿ vwd…u†, for all u; v; w A U. (iv) ‰u; vŠud…u† ˆ u‰u; vŠd…u†, forall u; v A U. ÿ (v) ‰u; vŠ d…uv† ÿ ud…v† ÿ vd…u† ˆ 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 † ˆ 2ud…u† for all u A U, then ‰u; vŠd…‰u; vŠ† ˆ 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 ˆ 0, then a ˆ 0 or b ˆ 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† ˆ u 2 d…v† ‡ …uv ‡ vu†d…u† ‡ ud…‰u; vŠ†, for all u; v A U. (ii) d…vu 2 † ˆ u 2 d…v† ‡ …3vu ÿ uv†d…u† ÿ ud…‰u; vŠ†, for all u; v A U. Proof. (i) Since uv ‡ vu ˆ …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 † ˆ 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† ˆ 2 ud…uv† ‡ uvd…u† ;

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for all u; v A U:

…2†

Now, subtracting equation (1) from equation (2) yields that ÿ  d…u 2 v ÿ vu 2 † ˆ 2 ud…‰u; vŠ† ‡ ‰u; vŠd…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 † ˆ 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† ˆ u 2 d…v† ‡ …uv ‡ vu†d…u† ‡ ud…‰u; vŠ†;

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 † ˆ u 2 d…v† ‡ …3vu ÿ uv†d…u† ÿ ud…‰u; vŠ†;

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† ˆ 0;

for all u; v A U:

…5†

Replacing u by ‰u; wŠ in (5), we get ‰u; wŠ 2 vd…‰u; wŠ† ÿ 2‰u; wŠv‰u; wŠd…‰u; wŠ† ‡ v‰u; wŠ 2 d…‰u; wŠ† ˆ 0;

for all u; v; w A U:

Now, application of Lemma 2 yields that ‰u; wŠ 2 Ud…‰u; wŠ† ˆ …0†. Hence by Lemma 3, either ‰u; wŠ 2 ˆ 0 or d…‰u; wŠ† ˆ 0. If ‰u; wŠ 2 ˆ 0, for all u; w A U, then linearizing the above relation on w, we get ‰u; wŠ‰u; vŠ ‡ ‰u; vŠ‰u; wŠ ˆ 0;

for all u; v; w A U:

…6†

For any u; v A U, uv ‡ vu ˆ …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; wŠ‰u; vŠu ‡ ‰u; vŠu‰u; wŠ ˆ 0;

for all u; v; w A U;

and hence by application of (6), we obtain ‰u; vŠ‰u; ‰u; wŠŠ ˆ 0. Now replace v by 2vv1 , to get ‰u; vŠv1 ‰u; ‰u; wŠŠ ˆ 0, for all u; v; v1 ; w A U i.e. ‰u; vŠU‰u; ‰u; wŠŠ ˆ …0†. Thus by Lemma 3, we ®nd that for each u A U, either ‰u; vŠ ˆ 0 or ‰u; ‰u; wŠŠ ˆ 0. If ‰u; ‰u; wŠŠ ˆ 0, for all w A U, then replacing w by 2vw, we get ‰u; vŠ‰u; wŠ ˆ 0, for all v; w A U. Again replace w by 2wv, to get ‰u; vŠw‰u; vŠ ˆ 0, for all v; w A U i.e. ‰u; vŠU‰u; vŠ ˆ …0†, for all v A U and hence by Lemma 3, we get ‰u; vŠ ˆ 0. Thus, in

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both the cases we ®nd that ‰u; vŠ ˆ 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; wŠ† ˆ 0, for all u; w A U. So we have ÿ  ÿ  2d …wu†u ˆ d …wu†u ‡ u…wu† ˆ 2fu 2 d…w† ‡ uwd…u† ‡ wud…u†g: Since R is a 2-torsion free, we obtain ÿ  d …wu†u ˆ u 2 d…w† ‡ uwd…u† ‡ wud…u† for all u; w A U:

…7†

Using Lemma 4(ii) and (7), we get ‰u; wŠd…u† ˆ 0;

for all u; w A U:

…8†

Now, replacing w by 2wv in (8) and since char R 0 2, we get ‰u; wŠvd…u† ˆ 0 i.e. ‰u; wŠUd…u† ˆ …0†. Thus, by Lemma 3, we ®nd that for each u A U either ‰u; wŠ ˆ 0 or d…u† ˆ 0. If ‰u; wŠ ˆ 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† ˆ 0 ie. d…U† ˆ …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).