International Journal of Algebra, Vol. 5, 2011, no. 1, 17 - 23
A Note on Generalized Derivations
of Prime Rings
Neşet AYDIN
Çanakkale Onsekiz Mart University
Faculty of Arts and Science
Department of Mathematics
17020 Çanakkale, Turkey
neseta@comu.edu.tr
Abstract
In the present paper we prove the following result; Let R be a noncommutative prime ring, I an ideal of R, (F, d) a generalized derivation
of R and a ∈ R. If F ([x, a]) = 0 or [F (x), a] = 0 for all x ∈ I, then,
d(x) = λ[x, a] for all x ∈ I or a ∈ Z.
Mathematics Subject Classification: 16N60, 16W25, 16U80
Keywords: Derivations, Generalized Derivations, Prime Rings, Ideals
1
Introduction
Throughout R will be a ring with center Z. Let x, y ∈ R. The commutator
xy − yx will be denoted by [x, y]. Recall that a ring is prime if xRy = 0 implies
x = 0 or y = 0. An additive mapping ∂ : R → R is called a derivation if
∂(xy) = ∂(x)y + x∂(y) holds for all x, y ∈ R. The study of commutativity
of prime rings with derivation was initiated by Posner [6]. During the past
few decades, there has been an ongoing interest concerning the relationship
between the commutativity of a ring and the existence of certain specific types
of derivations of R. Recently, in [2], Bresar defined the following notation. An
additive mapping F : R → R is called a generalized derivation if there exists
a derivation d : R → R such that F (xy) = F (x)y + xd(y), for all x, y ∈ R.
Throughout in this paper, R will be a prime ring with Martindale ring of
quotients Qr (R), extended centroid C and central closure RC = RC (see [5, sec.
10], for detail). In [3], B. Havala studied generalized derivation can be uniquely
extended to a generalized derivation of Qr (R). In [1], Albas, E. and Argac,
N. showed that R is a non-commutative ring and d a generalized derivation
18
Neşet AYDIN
determined by a derivation α of R, a ∈ R and for all x ∈ R, [a, d(x)] = 0 then
either a ∈ C or there exist λ, η ∈ C such that d(x) = ηx + λ(ax + xa), for
all x ∈ R. In this paper, our aim is to extend this result to a non-zero ideal
I of R. In particular, our research can be viewed as a new more elementary
approach.
2
Main Results
In the following, we assume that R is a prime ring. We denote a generalized
derivation F : R → R determined by a derivation d of R by (F, d).
Lemma 2.1 Let R be a non-commutative ring, I an ideal of R and (F, d)
a generalized derivation of R and a ∈ R. If a ∈
/ Z and [F (x), a] = 0 for all
x ∈ I, then F ([x, a]) = 0 for all x ∈ I.
Proof. We replace x by xr, r ∈ R in the defining equation
[F (x), a] = 0 for all x ∈ I
(1)
to obtain,
0 = [F (xr), a] = [F (x)r + xd(r), a]
= [F (x)r, a] + [xd(r), a]
= F (x)[r, a] + [F (x), a]r + x[d(r), a] + [x, a]d(r)
for all x ∈ I, r ∈ R which implies that
F (x)[r, a] + x[d(r), a] + [x, a]d(r) = 0
(2)
for all x ∈ I, r ∈ R. In (2), replace x by xy, y ∈ I and use (2) , we obtain
0 = F (xy)[r, a] + xy[d(r), a] + x[y, a]d(r) + [x, a]yd(r)
= F (x)y[r, a] + xd(y)[r, a] + xy[d(r), a] + x[y, a]d(r) + [x, a]yd(r)
= F (x)y[r, a] + xd(y)[r, a] + x(y[d(r), a] + [y, a]d(r)) + [x, a]yd(r)
= F (x)y[r, a] + xd(y)[r, a] − xF (y)[r, a] + [x, a]yd(r)
= (F (x)y + xd(y) − xF (y))[r, a] + [x, a]yd(r)
so we get
(F (x)y + xd(y) − xF (y))[r, a] + [x, a]yd(r) = 0, ∀x, y ∈ I, r ∈ R
Replace r by a in (3), we have
[x, a]Id(a) = 0, ∀x ∈ I
(3)
Generalized Derivations of Prime Rings
19
Since a ∈
/ Z and the primeness of I, yields
d(a) = 0
If we substitute sx, s ∈ R for x in (3), then we get
0 = (F (sx)y + sxd(y) − sxF (y))[r, a] + [sx, a]yd(r)
= ((F (s)x + sd(x))y + sxd(y) − sxF (y))[r, a] + s[x, a]yd(r) + [s, a]xyd(r)
= (F (s)xy + sd(x)y + sxd(y) − sxF (y))[r, a] + s[x, a]yd(r) + [s, a]xyd(r)
= F (s)xy[r, a] + sd(x)y[r, a] + sxd(y)[r, a] − sxF (y)[r, a]
+ s[x, a]yd(r) + [s, a]xyd(r)
= (F (s)xy + sd(x)y)[r, a] + s ((xd(y) − xF (y))[r, a] + [x, a]yd(r))
+ [s, a]xyd(r)
= (F (s)xy + sd(x)y)[r, a] + s(−F (x)y[r, a]) + [s, a]xyd(r)
= (F (s)xy + sd(x)y − sF (x)y)[r, a] + [s, a]xyd(r)
and so
(F (s)x + sd(x) − sF (x))y[r, a] + [s, a]xyd(r) = 0
(4)
In (4) replacing s by a,
(F (a)x + ad(x) − aF (x))y[r, a] = 0 ∀x, y ∈ I
(5)
Using a ∈
/ Z and the primeness of I, we obtain
F (a)x + ad(x) − aF (x) = 0.
Then we have
F (ax) = aF (x), ∀x ∈ I
(6)
On the other hand, by employing d(a) = 0, we see that the relation
F (xa) = F (x)a + xd(a) = F (x)a
is reduced to
F (xa) = F (x)a, ∀x ∈ I
Combining (6) and (7), we arrive at
F ([x, a]) = F (xa) − F (ax) = F (x)a − aF (x)
By using the hypothesis, we have
F ([x, a]) = [F (x), a] = 0, ∀x ∈ I
This completes the proof.
(7)
20
Neşet AYDIN
Lemma 2.2 Let R be a non-commutative prime ring, I an ideal of R, (F, d)
a generalized derivation of R and a ∈ R. If a ∈
/ Z and F ([x, a]) = 0 for all
x ∈ I, then [F (x), a] = 0 for all x ∈ I.
Proof. we replace x by xa in the defining equation F ([x, a]) = 0 to obtain
0 = F ([xa, a]) = F ([x, a]a)
= F ([x, a])a + [x, a]d(a)
and so
[x, a]d(a) = 0, for all x ∈ I
(8)
Taking xy, y ∈ I instead of x in (8), 0 = [xy, a]d(a) = x[y, a]d(a) + [x, a]yd(a)
and using (8) we obtain
[x, a]Id(a) = 0, for all x ∈ I
(9)
By the primeness of I and a ∈
/ Z, (9) implies that
d(a) = 0
Now we replace x by xy, y ∈ I in the defining equation F ([x, a]) = 0 to obtain
0 = F ([xy, a]) = F (x[y, a] + [x, a]y)
= F ([x, a]y) + F (x[y, a])
= F ([x, a])y + [x, a]d(y) + F (x)[y, a] + xd([y, a])
= [x, a]d(y) + F (x)[y, a] + x([d(y), a] + [y, d(a)])
Since d(a) = 0, we have
F (x)[y, a] + [x, a]d(y) + x[d(y), a] = 0, ∀x, y ∈ I
(10)
Substitute yz, z ∈ I instead of y in (10) and use (10), we arrive at
0 = F (x)[yz, a] + [x, a]d(yz) + x[d(yz), a]
= F (x)y[z, a] + F (x)[y, a]z + [x, a]d(y)z + [x, a]yd(z) + x[d(y)z, a] + x[yd(z), a]
= F (x)y[z, a] + (F (x)[y, a] + [x, a]d(y))z + [x, a]yd(z) + xd(y)[z, a]
+ x[d(y), a]z + xy[d(z), a] + x[y, a]d(z)
= F (x)y[z, a] + (F (x)[y, a] + [x, a]d(y) + x[d(y), a])z + [x, a]yd(z)
+ xd(y)[z, a] + xy[d(z), a] + x[y, a]d(z)
= F (x)y[z, a] + [x, a]yd(z) + xd(y)[z, a] + xy[d(z), a] + x[y, a]d(z)
= (F (x)y + xd(y))[z, a] + [x, a]yd(z) + x(y[d(z), a] + [y, a]d(z))
= (F (x)y + xd(y))[z, a] + [x, a]yd(z) − xF (y)[z, a]
Generalized Derivations of Prime Rings
21
and so
(F (x)y + xd(y) − xF (y))[z, a] + [x, a]yd(z) = 0, ∀x, y, z ∈ I
(11)
Replace x by ax in (11) and use (11), it yields
0 = (F (ax)y + axd(y) − axF (y))[z, a] + a[x, a]yd(z)
= F (ax)y[z, a] + a(xd(y)[z, a] − xF (y)[z, a] + [x, a]yd(z))
= F (ax)y[z, a] − aF (x)y[z, a]
Hence we get
(F (ax) − aF (x))y[z, a] = 0, ∀x, y, z ∈ I
(12)
Since a ∈
/ Z and the primeness of I, we have
F (ax) = aF (x), ∀x ∈ I
(13)
On the other hand, since d(a) = 0,
F (xa) = F (x)a + xd(a) = F (x)a
(14)
Combining (13) and (14) we arrive at
[F (x), a] = F (x)a − aF (x)
= F (xa) − F (ax)
= F ([x, a]) = 0
and so
[F (x), a] = 0, ∀x ∈ I
Thus. the proof is complete.
The following theorem is motivated by [1, Theorem 3.1 and Corollary 3.6].
Theorem 2.3 Let R be a non-commutative prime ring, I an ideal of R,
(F, d) a generalized derivation of R and a ∈ R. If a ∈
/ Z and F ([x, a]) = 0 or
[F (x), a] = 0 for all x ∈ I, then d(x) = λ[x, a], for all x ∈ I.
Proof. Since a ∈
/ Z and [F (x), a] = 0 for all x ∈ I, then by Lemma 2.1 we
have
F ([x, a]) = 0 and d(a) = 0
22
Neşet AYDIN
By the proof of the Lemma 2.1, we have the equation (3); in the equation (3),
replace y by [a, y] then we get
0 = (F (x)[a, y] + xd([a, y]) − xF ([a, y]))[r, a] + [x, a][a, y]d(r)
= (F (x)[a, y] + x[a, d(y)][r, a] + [x, a][a, y]d(r)
= −(F (x)[y, a] + x[d(y), a])[r, a] + [x, a][a, y]d(r)
In the above equation, using the (10) equation [a, x]d(y) = F (x)[y, a]+x[d(y), a]
in the proof of the Lemma 2.1, we obtain
[a, x](d(y)[r, a] − [y, a]d(r)) = 0
Define h : R → R, h(x) = [a, x], then the above equation yields h(x)(d(y)[r, a]−
[y, a]d(r)) = 0. Since a ∈
/ Z, by [6, Lemma 1] we get
d(y)[r, a] = [y, a]d(r), ∀y ∈ I, r ∈ R
(15)
Replace r by rs, s ∈ R in (15) and use (15), we obtain
d(y)r[s, a] = [y, a]rd(s), ∀r, s ∈ R, y ∈ I
(16)
Substitute yz, z ∈ R instead of y in (16) and use (16) it gives us
d(z)r[s, a] = [z, a]rd(s)
(17)
Now, define g : R → R, g(x) = [x, a], then from (17) we have
d(z)rg(s) = g(z)rd(s), ∀r, s, z ∈ R
Since g = 0, by [4, Lemma 1.3.2] for some λ ∈ C,
d(x) = λ[x, a]
Thus, the the proof is complete.
References
[1] Albas, E. and Argaç, N., Generalized Derivations of Prime Rings, Algebra
Colloquium, 11:3, (2004), 399-410.
[2] Bresar, M., On the distance of the composition of two derivations to the
generalized derivations, Glasgow Math. J., 33, (1991), 89-93.
Generalized Derivations of Prime Rings
23
[3] Havala, B., Generalized derivation in prime rings, Comm. Algebra, 26 (4),
(1998), 1147-1166.
[4] Herstein, I. N. Rings with Involution, Chicago Lectures in Mathematics,
The University of Chicago Press, Chicago, Ill.-London, 1976.
[5] Passman, D., Infinite Crossed Products, Academic Press, San Diego,1989.
[6] Posner, E. C. Derivations in prime rings, Proc Amer. Math. Soc., 8, (1957),
1093-1100.
Received: August, 2010