Derivations as Homomorphisms or Anti-homomorphisms on Lie Ideals

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Sep 21, 2006 - ([d(x),y]+[x, d(y)])[x1,y1]+[x, y]([d(x1),y1]+[x1,d(y1)]) ... for all x, y, x1,y1 ∈ I. If d is not inner on Q, applying Kharchenko's theorem in [4] we get.

Acta Mathematica Sinica, English Series Jun., 2007, Vol. 23, No. 6, pp. 1149–1152 Published online: Sep. 21, 2006 DOI: 10.1007/s10114-005-0840-x Http://www.ActaMath.com

Derivations as Homomorphisms or Anti-homomorphisms on Lie Ideals Yu WANG Department of Mathematics, Jilin Normal University, Siping 136000, P. R. China E-mail: ywang [email protected]

Hong YOU College of Sciences, Harbin Institute of Technology, Harbin 150001, P. R. China E-mail: [email protected] Abstract Let R be a 2-torsion free prime ring, Z the center of R, and U a nonzero Lie ideal of R. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U , then either d = 0 or U ⊆ Z. This result improves a theorem of Asma, Rehman, and Shakir. Keywords prime ring, lie ideal, derivation, homomorphism, anti-homomorphism MR(2000) Subject Classification 16N60, 16W25

1 Introduction Given a ring A and x, y ∈ A, the symbol [x, y] stands for the commutator xy − yx. A derivation d of A is a map from A into A satisfying d(x + y) = d(x) + d(y) and d(xy) = d(x)y + xd(y) for all x, y ∈ A. Let R be a prime ring with Z the center of R. In [1] Bell and Kappe proved that if d is a derivation of R which acts as a homomorphism or an anti-homomorphism on a nonzero right ideal I of R, then d = 0 on R. Recently Asma, Rehman and Shakir in [2] proved a similar result for Lie ideals. To be more specific, the statement of Asma, Rehman and Shakir’s theorem is the following Theorem 1.1 ([2, Theorem 2.2]) Let R be a 2-torsion free prime ring and U a nonzero Lie ideal of R such that u2 ∈ U , for all u ∈ U . If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U , then either d = 0 or U ⊆ Z. The goal of this paper is to eliminate the hypothesis, u2 ∈ U for all u ∈ U in Theorem 1.1. We will prove the following Theorem 1.2 Let R be a 2-torsion free prime ring and U a nonzero Lie ideal of R. If d is a derivation of R which acts as a homomorphism or an anti-homomorphism on U , then either d = 0 or U ⊆ Z. 2 Proof of Theorem 1.2 Throughout this section, let R be a 2-torsion free prime ring, Z the center of R, Q the maximal right quotient ring of R, C the extended centroid of R, and U a Lie ideal of R. All these notions are explained in detail in the book [3]. A derivation d of R can be extended uniquely to a derivation on Q (see [3, Proposition 2.5.1]) which will be also denoted by d. A derivation Received December 2, 2004, Accepted January 19, 2005 Partially supported by China Postdoctoral Science Foundation

Wang Y., You H.

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d is said to be Q-inner if there exists b ∈ Q such that d(x) = [b, x] for all x ∈ R, otherwise d is Q-outer. Moreover we remark that the main tool will be the theory of differential identities, initiated by Kharchenko in [4]. Proof Assume to the contrary that both d = 0 and U ⊆ Z. Since R ia a 2-torsion free prime ring, by a theorem of Herstein in [5], U ⊇ [I, R] = 0, for some I, a nonzero ideal of R. Without loss of generality we can assume U = [I, I]. We divide the proof into two cases. Case 1

If d acts as a homomorphism on U , then we have d([x, y])[x1 , y1 ] + [x, y]d([x1 , y1 ]) = d([x, y][x1 , y1 ]) = d([x, y])d([x1 , y1 ])

(1)

for all x, y, x1 , y1 ∈ I. Equivalently we have ([d(x), y] + [x, d(y)])[x1 , y1 ] + [x, y]([d(x1 ), y1 ] + [x1 , d(y1 )]) = ([d(x), y] + [x, d(y)])([d(x1 ), y1 ] + [x1 , d(y1 )]) for all x, y, x1 , y1 ∈ I. If d is not inner on Q, applying Kharchenko’s theorem in [4] we get ([s, y] + [x, t])[x1 , y1 ] + [x, y]([s1 , y1 ] + [x1 , t1 ]) = ([s, y] + [x, t])([s1 , y1 ] + [x1 , t1 ]) for all x, y, x1 , y1 , s, t, s1 , t1 ∈ I. In particular, for y = s1 = t1 = 0, [x, t][x1 , y1 ] = 0 for all x, x1 , y1 , t ∈ I. In other words, [I, I]2 = 0 and U 2 = 0. By [6, Lemma 4], U = 0, a contradiction. Let now d be an inner derivation induced by an element b ∈ Q, where b ∈ C. That is d(x) = [b, x] for all x ∈ R. Since by [3, Theorem 6.4.4] Q and I satisfy the same generalized polynomial identities (or GPIs in brief), it follows from (1) that [b, [x, y]][x1 , y1 ] + [x, y][b, [x1 , y1 ]] − [b, [x, y]][b, [x1 , y1 ]] = 0

(2)

for all x, y, x1 , y1 ∈ Q. In other words, Q satisfies the following identity: [b, [X, Y ]][X1 , Y1 ] + [X, Y ][b, [X1 , Y1 ]] − [b, [X, Y ]][b, [X1 , Y1 ]].

(3)

Since b ∈ C, the GPI (3) has the nonzero monomial bXY X1 Y1 . So it is a non-trivial generalized polynomial identity on Q.  In case C is infinite, GPI (3) is also satisfied  by Q C C where C is the algebraic closure of C (see [7, Proposition]). Since both Q and Q C C are prime  and centrally closed (see [8, Theorem 2.5 and Theorem 3.5]), we may replace R by Q or Q C C according as C is finite or infinite. Thus we may assume that R is centrally closed over Z which is either finite or algebraically closed and that b ∈ R, b ∈ Z such that R satisfies GPI (3). By Martindale’s theorem in [9], R is a primitive ring having nonzero socle and the commuting division D is a finite-dimensional central division algebra over Z. Since Z is either finite or algebraically closed, D must coincide with Z. Thus R is isomorphic to a dense subring of End(VZ ) for some vector space V over Z. For any given v ∈ V we claim that v and bv are Z-dependent. Suppose to the contrary that v and bv are Z-independent. By the density of R in End(VZ ) there exist elements x, y, x1 , y1 in R such that xv = 0, xbv = bv; yv = 0, ybv = v; x1 v = 0, x1 bv = v; y1 v = 0, y1 bv = bv. By (2) we have 0 = ([b, [x, y]][x1 , y1 ] + [x, y][b, [x1 , y1 ]] − [b, [x, y]][b, [x1 , y1 ]])v = v. This is a contradiction. Thus, v and bv are Z-dependent as claimed. From above we have proved that bv = vα(v) for all v ∈ V , where α(v) ∈ Z depends on v ∈ V . We claim that α(v) is independent of the choice of v ∈ V . Indeed, for any v, w ∈ V , if v and w are Z-independent, by the above situation, there exist α(v), α(w), α(v + w) ∈ Z such that

Derivations as Homomorphisms or Anti-homomorphisms

bv = vα(v), bw = wα(w), and

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b(v + w) = (v + w)α(v + w),

and so vα(v) + wα(w) = b(v + w) = (v + w)α(v + w). Hence v(α(v) − α(v + w)) + w(α(w) − α(v + w)) = 0. Since v and w are Z-independent, we have α(v) = α(v+w) = α(w). If v and w are Z-dependent, say v = wγ, where γ ∈ Z, then vα(v) = bv = bwγ = wα(w)γ = vα(w) and so α(v) = α(w) as claimed. So there exists δ ∈ Z such that bv = vδ for all v ∈ V . Hence b ∈ Z and so d = 0, a contradiction. Case 2

Now assume that d acts as an anti-homomorphism on U , so that d([x, y])[x1 , y1 ] + [x, y]d([x1 , y1 ]) = d([x, y][x1 , y1 ]) = d([x1 , y1 ])d([x, y])

(4)

for all x, y, x1 , y1 ∈ I. Expanding (4) we have ([d(x), y] + [x, d(y)])[x1 , y1 ] + [x, y]([d(x1 ), y1 ] + [x1 , d(y1 )]) = ([d(x1 ), y1 ] + [x1 , d(y1 )])([d(x), y] + [x, d(y)]) for all x, y, x1 , y1 ∈ I. If d is not inner on Q, applying Kharchenko’s theorem [4] we get ([s, y] + [x, t])[x1 , y1 ] + [x, y]([s1 , y1 ] + [x1 , t1 ]) = ([s1 , y1 ] + [x1 , t1 ])([s, y] + [x, t]) for all x, y, x1 , y1 , s, t, s1 , t1 ∈ I. In particular, for y = s1 = t1 = 0, [x, t][x1 , y1 ] = 0 for all x, x1 , y1 , t ∈ I. It follows from Case 1 that U = 0, a contradiction. Let now d be an inner derivation induced by an element b ∈ Q, where b ∈ C. That is d(x) = [b, x] for all x ∈ R. Since by [3, Theorem 6.4.4] Q and I satisfy the same generalized polynomial identities (or GPIs in brief), by (4) we have [b, [x, y]][x1 , y1 ] + [x, y][b, [x1 , y1 ]] − [b, [x1 , y1 ]][b, [x, y]] = 0

(5)

for all x, y, x1 , y1 ∈ Q. In view of the above situation in Case 1, we assume that R is centrally closed over Z which is either finite or algebraically closed and that b ∈ R, b ∈ Z such that R satisfies the non-trivial GPI (5). Moreover, we know that R is isomorphic to a dense subring of End(VZ ) for some vector space V over Z. For any given v ∈ V we claim that v and bv are Z-dependent. Suppose to the contrary that v and bv are Z-independent. By the density of R in End(VZ ) there exist elements x, y, x1 , y1 in R such that xv = 0, xbv = bv; yv = 0, ybv = v; x1 v = 0, x1 bv = v; y1 v = 0, y1 bv = bv. It follows from (5) that 0 = ([b, [x, y]][x1 , y1 ] + [x, y][b, [x1 , y1 ]] − [b, [x1 , y1 ]][b, [x, y]])v = v. This is a contradiction. Thus, v and bv are Z-dependent as claimed. In view of Case 1 we know that b ∈ Z and so d = 0, a contradiction. With this the proof is complete. References [1] Bell, H. E., Kappe, L. C.: Rings in which derivations satisfy certain algebraic conditions. Acta Math. Hungar., 53, 339–346 (1989) [2] Asma, A., Rehman, N., Shakir, A.: On Lie ideals with derivations as homomorphisms and anti-homomorphisms. Acta Math. Hungar., 101(1–2), 79–82 (2003) [3] Beidar, K. I., Martindale 3rd, W. S., Mikhalev, A. V.: Rings with Generalized Identities, Pure and applied Math., 196, Marcel Dekker, New York, 1996 [4] Kharchenko, V. K.: Differential identities of prime rings. Algebra and Logic, 17, 155–168 (1978) [5] Herstein, I. N.: Topics in Ring Theory, Univ. Chicago Press, Chicago, 1969 [6] Bergen, J., Herstein, I. N., Kerr, J. W.: Lie ideals and derivations of prime rings. J. Algebra, 71, 259–267 (1981)

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[7] Lee, P. H., Wong, T. L.: Derivations cocentralizing Lie ideals. Bull. Inst. Math. Acad. Sinica, 23, 1–5 (1995) [8] Erickson, J. S., Martindale 3rd, W. S., Osborn, J. M.: Prime nonassociative algebra. Pacific J. Math., 60, 49–63 (1975) [9] Martindale 3rd, W. S.: Prime rings satisfying a generalized polynomial identity. J. Algebra, 14, 576–584 (1969)

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