ALGEBRAS GROUPS M'D GEOMETRIES 25,175-180 (2008)
.
- 175 ­
ON ADDITIVE MAPPING IN SEMIPRIME RINGS
WITH LEFI'IDENTITY
Basudeb Dhara
Department of Mathematics
Belda College, Belda
Paschim Medinipur-721424 (W.B.), India
[email protected]
and
Rajendra K. Sharma
Department of Mat.lhematics
Indian Institute of Teclhnology, Delhi
Hauz Khas, New Delhi-l 10016, India
[email protected]
Revised July 27, 2008
Abstract
Let n > 2 be a fixed integer and let R be a (n + 1)1- torsion
free semiprime ring with left identity element. If D : R -+ R is an additive
mapping such that D(xn+l) = D(x)xn+xD(x)xn - 1 + ... +xnD(x) for all
x E R: then D is a derivation.
2000 Mathematics Subject Classification: 16\¥25, 16R50, 16N60.
J(ey words and phrases: Prime ring, semiprime ring, derivation.
o This work is S1J.pported by a grant from University Grants Commission, India.
Copyright © 2008 by Hadronic Press Inc., Palm Harbor, FL 34682, U.S.A.
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Throughout this paper, R denotes a associative ring with center Z(R).
The Lie commutator of .x, y is denoted by [x, y) and defined by [x,
xy - yx for x, y E R .
. A ring R is called n-torsion free if nx = 0 for x E R implies that x = O.
Recall that a ring R is prime ring if for a, b E R, aRh = (0) implies that
either a = 0 or b = 0, and is semiprime if aRa = (0) implies that a = O. An
additive mapping f : R - t R is said to be commuting on R if [I(x), = 0
for all x E R. An additive ma.pping D : R - t R is called a derivation if
D(xy) = D(x)y + xD(y) holds for all x,y E R, If an additive mapping D
from R to R satisfies D(x 2) = D(x)x+xD(x) for all x E R, then we call D as
a Jordan derivation. Obviously, every derivation is a Jordan derivation, but
the converse is not true in general. Herstein [3} has proved that every Jordan
derivation on a prime ring of characteristic different from 2 is a derivation.
Bresar and Vukman gave its brief proof in [I}. Cusack has generalized
Herstein's
result in [21. proving that every Jordan derivation on a 2-torsion
.
free semiprime ring is a derivation. In the present paper, our objective is to
generalize the previous identity by considering additive mapping with higer
power values of x. J\llore precisely, we prove the follovling:
Theorem 1. Let n 2: 2 be a fixed integer and let R be a (n + I)! - torsion
free semiprime ring with left identity element. If D : R ---+ R is an additive
mapping such that D(xn+l) = D(x)xn + xD(x)x n- 1 + ... + xnD(x) for all
x E R, then D is a derivation.
We need some identities which will be used to prove our main theorem.
(I) If e is a left identity in R, then for any x, y E R and any positive integer
k, .
(x
+ ke)n y
n+ (~)kxn-l + (;)k x n- + ...
+(n:l)kn - x + kne }y,
2
= {x
2
l
(l)
(H) If e is a left identity in R, then for any x E R and any positive integer
k,
(x + ke)n
=
x n + k{ (n~l)xn-l + xn-le}
+k n- 2 { (:::::;)x 2 +
(:::::~)x2e} + k
n
-
+ ... + kn - 3 { (:=;)x 3 + (:::::Dx 3 e}
1
{
(:::::i) x + (:::::;)xe} + kne.
(2)
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Proof of Theorem 1. "Ve are given that
D(xn+l) = D(x)xn
+ xD(x)x 11. - 1 +.:. + x n D(x)
for all x E R.
(3)
Let e be a left identity element in R. Then replacing x by e we have
nD(e)e = O. Since R is n-torsion free, we conclude that D(e)e = O. Now
since (e+ xe - x) is also a left identity in R, we can write D(e+xe - x)(e+
xe - x) = O. Since D(e)e = 0, it reduces to
D(xe - x)e + D(e)(xe - x)
+ D(xe - x)(xe - x) = O.
(4)
Now multiplying by e on the right, it gives
D(xe-x)e=O forallxER.
(5)
Replacing x by 2x in (4) and using 2-torsion freeness, we obtain
D(xe - x)e + D(e)(xe - x)
+ 2D(xe -
x)(xe - x) = O.
(6)
Now, (4) and (6) together implies D(xe-x)(xe-x) = 0 and hence D(xe­
x)e+D(e)(xe-x) = 0 for all x E R. Usi.."1g
it reduces to D(e)(xe-x) = 0
for all x E R. Now replacing x by xD(e) and using the fact D(e)e = 0, this
becomes D(e)xD(e) = 0 for all x E R, and then by semiprimeness of R we
have D(e) = O. Now, in (3) replace x by x + ke, where k is any positive
integer and -obtain
D«x + keY~+l) =
n-l
I: ex + ke)iD(x + ke)(x + ket- i + (x + ket D(x)
(7)
i=O
for all x E R. Expanding the power values of (x
D(e) = 0, we have
D[xn + 1
+ ke)
and using the fact
+ ... + kn- 2{ (n~2)x3 + (n~3)x3e}
+kn- 1{ (11.~1)x2 +
(11.~2)x2e} + kn{ (~)x + (n~l)xe}]
1: Ixi + ... + C~Jki-2X2 + (i~l)ki-lX + kie] D(x) [xn- i + ...
1
=
t-O _
+k11.-i-3{(n-~-1)x3
+ \11.-t-4
(11.-i-l)x3e} + k11.-i-2{(11.-~-1)x2
+ (n-~-1)x2e}
. 11.-t-3
11.-.-2
11.-t-3
+kn - i- 1{ (11.-i-l)
11.-t-1 x + (n-~-l)xe}
,11.-t-2
+ kn-ielJ
+{ xn + (~)kx11.-l +... + (n~2)kn-2x2 +(n~l)kn-lx + kne }D(X)
(8)
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for all x E R. Using relation (3), this can be re-written as
+ k 2h(x, e) + k 3hex, e) '+ ... + k n fn(x, e) = 0
kf1(X, e)
(9)
for all x E R. Now, replacing k by 1,2,3" .. , n in turn, and considering the
resulting system of n homogeneous equations, we see that the coefficient
matrix of the system is a Van der Monde matrix
Since the determinant of the matrLx is equal to a product of positive integers,
each of which is less than n, and since R is (n + I)!-torsion free, it follows
immediately that
flex, e) = fz(x, e) =
= 1.,. (x, e) =
...
O.
Now, fn(x, e) = 0 implies that
n-l
+ nD(xe) =
D(x)
L
D(x)e + D(x)
i=O
which gives nD(xe) = nD(x)e. Since R is n-torsion free, D{xe)
for all x E R. Again, fn-l(x, e) = 0 implies that
(/
11.
\D(x2)
,11.-1)
=
D(x)e
1
+ (n-2 )IJ(x2e) = nt
[D(X){· (n-~-1)x + (n-i-l)'
xe}
i = O . n-t-l·
n-t-2
11.
+(i~l)xD(x)e] + (n:l)xD(x)
for all x E R.
(10)
Multiplying both sides by 2, we have
2nD(x2 )
+ n(n -
[
1)D(x2 e) = 2 11.-1
i~oD(x)(x
+ (n - i - l)xe)
+ ixD(x)e]
+2nxD(x).
Using the fact D(xe)
2nD(x 2 )
= D(x)e for
+ n(n -
(11)
any x E R, it reduces to
I)D(x 2 )e
= 2nD(x)x + n(n - l)D(x)xe
+n(n - l)xD(x)e + 2nxD(x).
(12)
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Since R is n-torsion free,
2D(x2 )
+ (17,'-
1)D(x2 )e = 2D(x)x + (n - I)D(x)xe
+(n - l)xD(x)e + 2xD(x).
Right multiplying (13) by
C,
(13)
we get
(17, + I)D(x 2 )e = (17, + l)D(x)xe + (17, + I)xD(x)e.
(14)
Since R is (17, + I)-torsion free,
D(x2 )e = xD(x)e + D(x)xe for all x
E
R.
(15)
Using this and restriction of torsion freeness, (13) becomes,
D(x2 )
= xD(x) + D(x)x
for all x E R
(16)
that is D is a Jordan derivation. We recall that any Jordan derivation on a
2-torsion free semiprime ring is a derivation. Thus the proof of the theorem
is complete.
References
[1] M. Bresar and J. Vukman , Jordan derivations on prime rings, BuLL
Aust. Math. Soc. 37(1988) 321-324.
[2] J. M. Cusack, Jordan derivations on rings, Froc. Amer. Math. Soc.
53(1975)(2) 321-324.
[3] 1. N. Herstein, Jordan derivatio~ of prime rings, Proc. Amer. Math.
Soc. 8(1957) 1104-1110.