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On the annihilators of derivations with engel conditions in prime rings.

1. Introduction

Let R be an associative ring and Z(R) be its center. Let n be a positive integer. For x, y [member of] R, set [[x, y].sub.0] = x, [[x, y].sub.1] = [x, y] = xy . yx, then an Engel condition is a polynomial [[x, y].sub.k] = [[[x, y].sub.k-1], y], k = 1, 2, ... in noncommuting indeterminates.

A well known result of Posner (19) states that for a non-zero derivation d of a prime ring R, if [[d(x), x], y] = 0 for all x, y [member of] R, then R is commutative. In (16), Lanski generalized this result of Posner to the Lie ideal. Lanski proved that if U is a noncommutative Lie ideal of a prime ring R and d 6= 0 is a derivation of R such that [[d(x), x], y] = 0 for all x [member of] U, y [member of] R, then either R is commutative, or char R = 2 and R satises [S.sub.4], the standard identity in four variables. Bell and Martindale (4) studied this identity for a non-zero left ideal of R. They proved that if R is a semiprime ring and d a non-zero derivation such that [[d(x), x], y] = 0 for all x in a non-zero left ideal of R and y [member of] R, then R contains a non-zero central ideal. Clearly, this result says that if R is a prime ring, then R must be commutative.

Several authors have studied this kind of Engel type identities with derivation in dierent ways. In (11), Herstein proved that if R is a prime ring with char R [not equal to] 2 and R admits a non-zero derivation d such that [d(x), d(y)] = 0 for all x, y [member of] R, then R is commutative. In (10), Filippis showed that if R be a prime ring of characteristic dierent from 2, d a non-zero derivation of R and [rho] a non-zero right ideal of R such that [[rho], [rho]][rho] [not equal to] 0 and [[d(x), x], [d(y), y]] = 0 for all x, y [member of] [rho], then d([rho])[rho] = 0.

In the present paper we study this identity with annihilator conditions on prime rings in more generalized form.

Throughout this paper, unless specially stated, R always denotes a prime ring with center Z(R), with extended centroid C, and with two-sided Martin- dale quotient ring Q.

It is well known that any derivation of R can be uniquely extended to a derivation of Q, and so any derivation of R can be dened on the whole of Q. Moreover Q is a prime ring as well as R and the extended centroid C of R coincides with the center of Q. We refer to (2, 17) for more details.

Denote by Q*[.sub.C]C{X, Y}, Y g the free product of the C-algebra Q and C{X, Y}the free C-algebra in noncommuting indeterminates X, Y.

2. Main Results

We need the following lemma.

Lemma 2.1. Let [rho] be a non-zero right ideal of R and d a derivation of R. Then the following conditions are equivalent: (i) d is an inner derivation induced by some b [member of] Q such that [b.sub.[rho]][rho] = 0, (ii) d([rho])[rho] = 0 (For its proof we refer to [5, Lemma]).

We mention a important result which will be used quite frequently as follows:

Theorem (Kharchenko (14)): Let R be a prime ring, d a derivation on R and I a non-zero ideal of R. If I satisfies the differential identity

f([r.sub.1], [r.sub.2], ..., [r.sub.n], d([r.sub.1]), d([r.sub.2]), ... d([r.sub.n])) = 0 for any [r.sub.1], [r.sub.2], ... [r.sub.n] [member of] I

then either

(i)I satisfies the generalized polynomial identity

f([r.sub.1], [r.sub.2], ..., [r.sub.n], [x.sub.1]. [x.sub.2], ... [x.sub.n]) = 0

or (ii) d is Q-inner i.e., for some q [member of] Q, d(x) = [q, x] and I satisfies the generalized polynomial identity

f([r.sub.1], [r.sub.2], ..., [r.sub.n], [q, [r.sub.1]], [q, [r.sub.2]],..., [q, [r.sub.n]]) = 0.

Theorem 2.2. Let R be a prime ring of char R [not equal to] 2 and d a non-zero derivation of R and 0 [not equal to] b [member of] R such that b[[[d(x), x].sub.n], [[y, d(y)].sub.m]] = 0 for all x, y [member of] R, where n,m [greater than or equal to] 0 are fixed integers, then R is commutative.

Proof. If R is commutative, we have nothing to prove. So, let R be noncommutative. Assume first that d is Q-inner derivation, say d = ad(a) for some a [member of] Q i.e., d(x) = [a, x] for all x [member of] R. Then we have

b[[[a, x].sub.n+1], [y, [[a, y]].sub.m]] = 0

for all x, y [member of] R. Since d [not equal to] 0, a [??] C and hence R satisfies a nontrivial gener- alized polynomial identity (GPI). Since Q and R satisfy the same generalized polynomial identities with coeffcients in Q (6), f(x, y) = b[[[a, x].sub.n+1], [[y, [[a, y]].sub.m]] is also satisfied by Q. In case the center C of Q is infinite, we have f(x, y) = 0 for all x, y [member of] Q [[cross product].sub.C] [bar.C], where [bar.C] is the algebraic closure of C. Since both Q and Q [[cross product].sub.C] [bar.C] are prime and centrally closed [7, Theorem 2.5 and 3.5], we may replace R by Q or Q [[cross product].sub.C] [bar.C] according to C finite or infinite. Thus we may assume that R is centrally closed over C which is either finite or algebraically closed and f(x, y) = 0 for all x, y [member of] R. By Martindale's theorem (18), R is then a primitive ring having nonzero socle H with C as the associated division ring. Hence by Jacobson's theorem [13, p.75] R is isomorphic to a dense ring of linear transformations of some vector space V over C, and H consists of the linear transformations in R of finite rank. If V is a finite dimensional over C then the density of R on V implies that R [congruent to] [M.sub.k](C) where [kappa] = [dim.sub.C]V. We may assume that for some v [member of] V, {av, v} are linearly C-independent, for otherwise av - [alpha]v = 0 for all v [member of] V, that is (a - [alpha])V = 0 implying a = [alpha] [member of] C, a contradiction. If [a.sup.2]v [??] [span.sub.C]{v, av}, then {v, av, [a.sup.2]v} are all linearly C-independent. By density there exist x, y [member of] R such that xv = v, xav = 0, x[a.sup.2]v = 0, yv = 0, yav = v, y[a.sup.2]v = 0 for which we get

0 = b[[[a, x].sub.n+1], [[y, [a, y]].sub.m]]v = [-2.sup.m]bv.

If [a.sup.2]v [member of] [span.sub.C]{v, av}, then [a.sup.2]v = v[alpha] + av[beta]. Then again by density there exist x, y [member of] R such that xv = v, xav = 0, yv = 0, yav = v for which we get

0 = b[[[a, x].sub.n+1], [[y, [a, y]].sub.m]]v = [-2.sup.m]bv.

Thus in both the cases, whether [a.sup.2]v [??] [span.sub.C]{v, av}, we have that bv = 0, since char R [not equal to] 2. So, if for some v [member of] V, bv [not equal to] 0, then {v, av} be linearly C-dependent. Let bv = 0. Since b [not equal to] 0, there exists w [member of] V such that bw [not equal to] 0 and then b(v + w) = bw [not equal to] 0. Hence we have that {w. aw} are linearly C-dependent and {(v + w), a(v + w)g too. Thus there exist [alpha], [beta] [member of] C such that aw = w[alpha] = and a(v + w) = (v + w)[beta]. Moreover, v and w are clearly C-independent and so by density there exist x, y [member of] R such that xw = w, xv = 0, yw = v, yv = 0: Then we obtain by using bv = 0 that

0 = b[[[a, x].sub.n+1], [[y, [a, y]].sub.m]]w = [(-1).sup.n+1][2.sup.m]bw[([beta] - [alpha]).sup.3].

Since bw [not equal to] 0, [alpha] = [beta] and so av = v[alpha] contradicting the independency of v and av. Hence for each v [member of] V, [[alpha].sub.v] [member of] V, av = v[[alpha].sub.v] for some [[alpha].sub.v] [member of] C.It is very easy to prove that [[alpha].sub.v] is independent of the choice of v [member of] V. Thus we can write av = v[alpha] for all v V and [alpha] [member of] C fixed.

Now let r [member of] R, v [member of] V. Since av = v[alpha],

[a, r]v = (ar)v = (ra)v = a(rv) - r(av) = (rv)[alpha] - r(v[alpha]) = 0.

Thus [a, r]v = 0 for all v [member of] V i.e., [a, r]V = 0. Since [a, r] acts faithfully as a linear transformation on the vector space V, [a, r] = 0 for all r [member of] R. Therefore a [member of] Z(R) implies d = 0, ending the proof of this part.

Assume next that d is not Q-inner derivation in R. Then by Kharchenko's theorem (14), we have

b[[[u, x].sub.n], [[y, v].sub.m]] = 0

for all x, y, u, v [member of] R. Choose a [??] C. Then replacing u with [a, x] and v with [a, y], we obtain b[[[a, x].sub.n+1], [[y, [a, y]].sub.m]] = 0 for all x, y [member of] R, implying a [member of] C by same argument as earlier, a contradiction.

Theorem 2.3. Let R be a prime ring of char R [not equal to] 2, d a non-zero derivation of R and [rho] a non-zero right ideal of R such that b[[[d(x), x].sub.n], [[y, d(y)].sub.m]] = 0 for all x, y [member of] [rho], where n, m [greater than or equal to] 0 are fixed integers. If [[rho], [rho]][rho] [not equal to] 0, then either [b.sub.[rho] = 0 or d([rho])[rho] = 0.

We begin the proof by proving the following lemma

Lemma 2.4. Let [rho] be a nonzero right ideal of R, d a nonzero derivation of R and 0 [not equal to] b [member of] R such that b[[[d(x), x].sub.n], [[y, d(y)].sub.m]] = 0 for all x, y [member of] [rho] where n, m [greater than or equal to] 0 are fixed integers. Then if d([rho])[rho] [not equal to] 0 and [b.sub.[rho]] [not equal to] 0, R satisfies nontrivial generalized polynomial identity (GPI).

Proof. Suppose that d([rho])[rho] [not equal to] 0 and [b.sub.[rho] [not equal to] 0. Now we prove that R satisfies nontrivial generalized polynomial identity. On contrary, we assume that R does not satisfy any nontrivial GPI. We consider two cases

Case I. Suppose that d is an Q-inner derivation induced by an element a [member of] Q. Then for any x [member of] [rho]

b[[[[a,xX].sub.n+1], [[yY, [a, yY]].sub.m]]

is a GPI for R, so it is the zero element in Q *[.sub.C] C{X, Y}. Expanding this we get,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let ay and y are linearly C-independent for some y [member of] [rho]. Then a [??] C. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in Q *[.sub.C] C{X, Y} and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Again, since ay and y are linearly C-independent,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular,

b[[a, xX].sub.[n + 1]]yY[[( - yYa)].sup.m] = 0 (2.1)

that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since ay and y are linearly C-independent,

b[(-1).sup.n+1][(xX).sup.n+1] ayY [[( - yYa)].sup.m] = 0

in Q *[.sub.C] C{X, Y}. This implies bx = 0 for all x [member of] [rho] that is b[rho] = 0, a contradiction. Thus for any y [member of] [rho], ay and y are linearly C-dependent. Then (a - [alpha])[rho] = 0 for some [alpha] [member of] C. Replacing a with a - [alpha], we may assume that a[rho] = 0. Then by Lemma 2.1, d([rho])[rho] = 0, a contradiction.

Case II. Suppose that d is not Q-inner derivation. If for all x [member of] [rho], d(x) [member of] xC, then [d(x), x] = 0 which implies that R is commutative (see (3)). There-fore there exists x [member of] [rho] such that d(x) [??] xC i.e., x and d(x) are linearly C-independent.

By our assumption we have that R satisfies

b[[[d(xX), xX].sub.n], [[xY, d(xY)].sub.m]] = 0.

By Kharchenko's theorem (14),

b[[[d(x)X + xr1,xX].sub.n], [[xY, d(x)Y + xr2].sub.m]] = 0

for all X, Y, [r.sub.1], [r.sub.2] [member of] R. In particular for [r.sub.1] = [r.sub.2] = 0,

b[[[d(x)X,xX].sub.n], [[xY, d(x)Y].sub.m]] = 0

which is a non-trivial GPI for R, because x and d(x) are linearly C-independent, a contradiction.

We are now ready to prove our main Theorem.

Proof of Theorem 2.3. Suppose that d([rho])[rho] [not equal to] 0 and then we derive a contradiction. By Lemma 2.4, R is a prime GPI-ring, so is also Q by (6). Since Q is centrally closed over C, it follows from (18) that Q is a primitive ring with H = Soc(Q) [not equal to] 0.

By our assumption and by (17), we may assume that

b[[[d(x), x].sub.n], [[y, d(y)].sub.m]] = 0 (2.3)

is satisfied by [rho]Q and hence by [rho]H. Let e = [e.sup.2] [member of] H and y [member of] H. Then replacing x with e and y with ey(1 - e) in (2.3) and then right multiplying it by e we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we have the fact that for any idempotent e, d(y(1 - e))e = -y(1 - e)d(e), ed(e)e = 0 and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now since for any idempotent e and for any y [member of] R, (1 - e)d(ey) = (1 - e)d(e)y, above relation gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all y [member of] H. Since char R [not equal to] 2, we have by (9), Theorem 2] that bey(1 - e)d(e)e = 0 for all y [member of] H. By primeness of H, be = 0 or (1 - e)d(e)e = 0. By [8, Lemma 1], since H is a regular ring, for each r [member of] H, there exists an idempotent e [member of] [rho]H such that r = er and e [member of] rH. Hence be = 0 gives br = ber = 0 and (1 - e)d(e)e = 0 gives (1 - e)d(e) = (1 - e)d([e.sup.2]) = (1 - e)d(e)e = 0 and so d(e) = ed(e) [member of] [rho]H [??] [rho]H and d(r) = d(er) = d(e)er + ed(er) 2[member of] [rho]H. Hence for each r [member of] [rho]H, either br = 0 or d(r) [member of] [rho]H. Thus [rho]H is the union of its two additive subgroups {r [member of] [rho]H|br = 0} and {r [member of] [rho]H|d(r) [member of] [rho]H}. Hence b[rho]H = 0 and d([rho]H) [??] [rho]H. The case b[rho]H = 0 gives b[rho] = 0, a contradiction. Thus d([rho]H) [??] [rho]H. Set J = [rho]H. Replacing b with a nonzero element in Jb, we may assume that b [member of] J. Then [bar.J] = J/J[intersection [l.sub.H](J), a prime C-algebra with the derivation d such that d(x) = d(x), for all x [member of] J. By assumption we have that

[bar.b][[bar.d]([bar.x]),[bar.x]].sub.n, [[[bar.y].[bar.d]([bar.y]).sub.m] = 0

for all [bar.x], [bar.y] [member of] [bar.J]. By Theorem 2.2, we have either [bar.d] = 0, [bar.b] = 0, [bar.[rho]H] is commutative. Therefore we have that either d([rho]H)[rho]H = 0, b[rho]H = 0 or [[rho]H, [rho]H][rho]H = 0. Now d([rho]H)[rho]H = 0 implies 0 = d([rho][rho]H)[rho]H = d([rho])[rho]H[rho]H and so d([rho])[rho] = 0. b[rho]H = 0 implies b[rho] = 0. [[rho]H, [rho]H][rho]H = 0 implies 0 = [[rho][rho]H, [rho]H][rho]H = [[rho], [rho]H][rho]H[rho]H and so [[rho], [rho]H][rho] = 0 and then 0 = [[rho], [rho][rho]H][rho] = [[rho], [rho]][rho]H[rho] implying [[rho], [rho]][rho] = 0. Thus in all the cases we have contradiction. This completes the proof of the theorem.

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B. Dhara [dagger]

Department of Mathematics, Belda College, Belda, Paschim Medinipur-721424(W.B.), India

and

R. K. Sharma [double dagger]

Department of Mathematics, Indian Institute of Technology, Delhi,Hauz Khas, New Delhi-110016, India

Received May 20, 2008, Accepted September 28, 2009.

* 2000 Mathematics Subject Classification. Primary 16W25, 16R50, 16N60.

[dagger] Corresponding author. E-mail: basu_dhara@yahoo.com

[double dagger] E-mail: rksharma@maths.iitd.ac.in

On the Annihilators of Derivations with Engel Conditions in Prime Rings *
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Date:Dec 15, 2010
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