A note on generalized derivations on prime rings
A note on generalized derivations on prime rings
Maja Fošner 0
Nadeem ur Rehman 0
Tarannum Bano 0
Mathematics Subject Classification 0
0 N. ur Rehman
Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring Qs ( R). In this paper we prove the following result. Let F : R → R be a generalized derivation associated with a nonzero derivation d on R and let h be an additive map of R such that F (x )x = x h(x ) for all x ∈ R. Then either R is commutative or F (x ) = x p and h(x ) = px where p ∈ Qs ( R). Throughout the paper, R will be an associative ring with center Z . Recall that R is prime if for any a, b ∈ R, a Rb = 0 implies that a = 0 or b = 0. By Ql ( R) and Qr ( R) we denote the left Martindale ring of quotients of R and the right Martindale ring of quotients of R, respectively. Further, we denote by Qs ( R) the symmetric Martindale quotients ring of ring R. The center C of Qs ( R) is a field and it is the center of both Ql ( R) and Qr ( R). It is called the extended centroid of R. Also it is easily seen that C is the centralizer of R in both Qr ( R) and Ql ( R). In particular, Z ⊆ C . The subring of Qr ( R) (or Ql ( R)) generated by R and C is called the central closure of R and will be denoted by RC . Another subring of Qr ( R) is Qs ( R) = {q ∈ Qr I q ⊆ R

1 Introduction
for some nonzero ideal I of R}. It is called the symmetric Martindale ring of quotients. We point out that
R ⊆ RC ⊆ Qs (R) ⊆ Qr (R). Note that q1 Rq2 = 0, where q1, q2 ∈ Ql (R) or q1, q2 ∈ Qr (R) implies that
q1 = 0 or q2 = 0. In particular, this shows that all RC , Qs (R), Ql (R), and Qr (R) are prime rings, so that
one can construct (left, right, symmetric) Martindale ring of quotients and the central closure of each of these
rings.
Let RC ∗C C {X } be the free product over C of RC and the free algebra over C on an infinite set X of
indeterminates. A typical element in RC ∗C C {X } is a sum of monomials of the form λai0 x j1 ai1 x j2 . . . x jn ain
where λ ∈ C, aik ∈ RC and x jk ∈ X . R satisfy a generalized polynomial identity over C (simply R is a GPI
ring) if there exists a nonzero polynomial p(x1, x2, . . . , xn) ∈ RC ∗C C {X } such that p(r1, r2, . . . , rn ) = 0
for all r1, r2, . . . , rn ∈ R. We refer the reader to [2,3] for more details.
An additive map d : R → R is called a derivation if d(x y) = d(x )y + x d(y) holds for all x , y ∈ R. In
particular, for a fixed a ∈ R, the map Ia : R → R given by Ia (x ) = [x , a] is a derivation called an inner
derivation. Let S be a nonempty subset of R. A map f : R → R is said to be centralizing on S if [ f (x ), x ] ∈ Z
for every x ∈ S. In special case where [ f (x ), x ] = 0 for every x ∈ S, the map f is called commuting on S.
The study of centralizing maps was initiated by a wellknown theorem of Posner [15] which states that the
existence of a nonzero centralizing derivation on a prime ring R implies that R is commutative. A number of
authors have extended Posner’s theorem in several ways. They have showed that nonzero derivations cannot be
centralizing on various subsets of noncommutative prime rings (see [12] for probably the most general results
of the kind), and similar conclusion hold for some other maps. In [5] Brešar studied maps that are centralizing
and additive, and no further assumption was required. The main result of [5] characterizes commuting additive
maps on prime rings R: every such map is of the form x → λx + h(x ) where λ ∈ C , and h is an additive
map of R into C . Later, Lanski [13] dealt with the situation where a nonzero derivation d of a prime ring
R satisfies c1x d(y) + c2d(x )y + c3 yd(x ) + c4d(y)x ∈ C for some ci ∈ C and all x , y ∈ S, where S is a
subset of R. Neglecting rings of characteristic 2, the conclusion was: either all ci = 0 or R satisfies S4, the
standard identity of degree 4 (however, the exact statements are much more precise). The condition considered
by Lanski clearly covers the case of centralizing derivations, namely a linearization of [d(x ), x ] ∈ Z gives
x d(y)−d(x )y + yd(x )−d(y)x ∈ Z . The same is true for the case skewcentralizing on S if f (x )x +x f (x ) ∈ Z
for all x ∈ S. In the special case where f (x )x + x f (x ) = 0 for all x ∈ S, the map f is called skewcommuting
on S. In [6] Brešar proved that there is no nonzero additive maps that are skewcommuting on ideals of prime
rings of characteristic not 2.
An additive map F : R → R is called a generalized inner derivation if F (x ) = ax + x b for fixed a, b ∈ R.
For such a map F , it is easy to see that F (x y) = F (x )y + x [y, b] = F (x )y + x Ib(y) for all x , y ∈ R. This
observation leads to the following definition, given in [4] and [9]; an additive map F : R → R is called a
generalized derivation with associated derivation d if F (x y) = F (x )y + x d(y) holds for all x , y ∈ R.
Familiar examples of generalized derivations are derivations and generalized inner derivations and the later
includes left multiplier, i.e., an additive map F : R −→ R satisfying F (x y) = F (x )y for all x , y ∈ R. In
[11], Lee extended the definition of a generalized derivation as follows: by a generalized derivation we mean an
additive mapping F : I → U such that F (x y) = F (x )y + x d(y) holds for all x , y ∈ I , where I is a dense left
ideal of R, U is the Utumi quotient ring (i.e., the maximal right quotient ring) of R and d is a derivation from
I to U . Moreover, Lee also proved that every generalized derivation can be uniquely extended to a generalized
derivation of U , and thus all generalized derivations of R will implicitly assumed to be defined on the whole
of U . Lee obtained the following: every generalized derivation F on a dense left ideal of R can be extended
to U and assumes the form F (x ) = ax + d(x ) for some a ∈ U and a derivation d on U .
Motivated by the work of Brešar [5] and Lanski [13], in this paper we consider F as a generalized derivation
and h as an additive map of R such that F (x )x = x h(x ) for all x ∈ R. In fact, our result extends Posner’s
Theorem [15], Brešar’s Theorem [5] and Ashraf et al. [1].
2 Main result
In this section we shall prove our main theorem. Before that we need some known results:
Lemma 2.1 [14, Theorem 3] A prime ring R satisfies a GPI if and only if RC is a primitive ring with nonzero
socle and e RC e is a finitedimensional division algebra over C for each primitive idempotent e in RC .
Lemma 2.2 [7, Main Theorem] Let R be prime ring and let n, m, k, l be positive integers. Suppose that
in=1 Fi (y)x ai +
m
i=1Gi (x )ybi +
ik=1ci y Hi (x ) +
l
i=1di x Ki (y) = 0
for all x, y ∈ R, where Fi , Gi , Hi , Ki : R → RC are additive maps and {a1, . . . , an}, {b1, . . . , bm}, {c1, . . . ,
ck }, {d1, . . . , dl } are Cindependent subsets of R. Then one of the two possibilities holds:
(i) RC is a primitive ring with nonzero socle and e RC e is a finitedimensional division algebra over C for
each primitive idempotent e in RC ( that is, R is a GPI ring),
(ii) There exists elements qi j ∈ Qs (RC ), i = 1, . . . , l, j = 1, . . . , m, pi j ∈ Qs (RC ), i = 1, . . . , k, j =
1, . . . , n and additive maps λi j : R → C, i = 1, . . . , l, j = 1, . . . , n, μi j : R → C, i = 1, . . . , m, j =
1, . . . , k, such that
Fi (y) =
kj=1c j yp ji + lj=1λ ji (y)d j , for all y ∈ R, i = 1, . . . , n,
Gi (x) = lj=1d j xq ji − kj=1μi j (x)c j , for all x ∈ R, i = 1, . . . , m,
Hi (x) = − nj=1 pi j xa j + mj=1μ ji (x)b j , for all x ∈ R, i = 1, . . . , k,
Ki (y) = − mj=1qi j yb j − lj=1λ ji (y)a j , for all y ∈ R, i = 1, . . . , l.
Lemma 2.3 [10, Theorem 2] Let R be a prime ring, U its maximal right quotient ring and IR a dense
Rsubmodule of UR. Then I and U satisfy the same differential identities
Now we are well equipped to prove our theorem:
Theorem 2.4 Let R be a prime ring and F : R → R be a generalized derivation associated with a nonzero
derivation d. Further let h be an additive map of R such that F(x)x = x h(x) for all x ∈ R. Then either R is
commutative or F(x) = x p and h(x) = px where p ∈ U .
Proof We have F(x)x = x h(x) for all x ∈ R. Linearizing this relation we have
F(y)x + F(x)y − yh(x) − x h(y) = 0
for all x, y ∈ R. We solve this functional identity in two different cases.
Case I: R is not a GPI ring. Using Lemma 2.2, we get from (
1
)
where λ, μ : R → C additive maps. From (
2
), we have F(y) − yp = λ(y) ∈ C. Let G be the additive map
defined as G(y) = F(y) − yp for any element y ∈ R. Since F is a generalized derivation with associated
derivation d, first we prove that G is a generalized derivation of R.
G(x y) = F(x y) − x yp = F(x)y + xd(y) − x yp
= F(x)y + xd(y) − x yp + (x py − x py)
= (F(x) − x p)y + x(d(y) + [ p, y])
= G(x)y + xg(y),
where g(x) = d(x) + [ p, x] is the associated derivation of G. Hence G is a generalized derivation. Thus, by
(
2
) G(y) is central in R, for any element y ∈ R. Hence by Hvala [9, Lemma 3] either R is commutative or
G = 0, which imply F(y) − yp = 0 and hence F(y) = yp for any y ∈ R. Similarly from (
3
), we find that
either R is commutative or F(x) = xq. These two relations imply that p = q and λ = μ = 0 and hence
h(x) = px where p ∈ U .
Case II: R is a GPI ring. If there exists a nonzero idempotent e in RC . If there exists e2 = e = {0, 1} in
Qs (R). Therefore, we can find a nonzero ideal I of R satisfying eI + I e ⊆ R. Then from (
1
), we get
F(ey)ex + F(ex)ey = ex h(ey) + eyh(ex), for all x, y ∈ I.
Thus
F(ey)ex + F(ex)ey = e{F(ey)ex + F(ex)ey}, for all x, y ∈ I.
F(y) = yp + λ(y),
F(x) = xq + μ(x),
h(x) = px + μ(x),
h(y) = q y + λ(y),
(
1
)
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
This implies that
By Lemma 2.3, Qs (R) and I satisfy the same differential identity. Thus we have
(1 − e)F (ey)ex + (1 − e)F (ex )ey = 0, for all x , y ∈ I.
(1 − e)F (ey)ex + (1 − e)F (ex )ey = 0, for all x , y ∈ Qs (R).
where H (x ) = (1 − e)F (ex )e. Replacing x by x z in (
6
), we get
Multiplying (
6
) by z from the right, we find that
Comparing (
7
) and (
8
), we find that
Replacing y by yu in (
9
), we get
Right multiplication of (
9
) by u, we get
H (x )y + H (y)x = 0,
H (x z)y + H (y)x z = 0.
H (x )yz + H (y)x z = 0.
H (x z)y = H (x )yz.
H (x z)yu = H (x )yuz.
H (x z)yu = H (x )yzu.
Comparing (
10
) and (
11
), we get H (x )y[z, u] = 0 for all x , y, z, u ∈ Qs (R). Since R is prime and Qs (R) is
also prime, we get from the last relation either R is commutative or H (x ) = 0 for all x ∈ Qs (R). If H (x ) = 0,
we get (1 − e)F (ex )e = 0 and considering F (x ) = ax + d(x ) for a ∈ U , we find that
This implies
(1 − e)[aex + d(ex )]e = 0.
(1 − e)aex e + (1 − e)d(e)x e = 0.
Let x y−1 = u in (
12
), we get
Since R is prime and e is nontrivial, we get (1 − e)ae + (1 − e)d(e) = 0 for all nontrivial idempotents
e ∈ Qs (R). Replacing e by 1 − e we get ea(1 − e) + ed(−e) = 0. Combining these two relations we get
d(e)+[a, e] = 0. Let E be an additive subgroup generated by idempotents in Qs (R). Therefore, d(u)+[a, u] =
0 for all u ∈ E . Now, for all u, v ∈ E , we get d(uv) + [a, uv] = 0 = (d(u) + [a, u])v + u(d(v) + [a, v]).
That is, d(u) + [a, u] = 0 for all u ∈ E = [E , E ]. Now [E , E ] = 0, since [e, e + ex (1 − e)] = 0 for some
x ∈ Qs (R). By Herstein’s arguments [8, page 4] 0 = Qs [E , E ]Qs ⊆ E , W = Qs [E , E ]Qs is a nonzero
ideal of Qs . Therefore, d(u) + [a, u] = 0 for all u ∈ W and hence d(x ) + [a, x ] = 0 for all x ∈ Qs (R) by
Lemma 2.3. Thus F (x ) = ax + d(x ) = ax − [a, x ] = x a.
By Martindale’s theorem [14, Theorem 3] we know that RC is a primitive ring and H = soc(RC ) = 0 and
e RC e is a finite dimensional for any minimal idempotent e. If H contains no nontrivial idempotent, then H
is a finitedimensional division algebra over C . If soc (RC ) contains no nontrivial idempotent, then soc (RC )
must be a finitedimensional division algebra over C , by [14, Theorem 3]. Since soc (RC ) is a nonzero ideal
of RC , it follows RC = soc (RC ) is a division algebra. Then for x = 0 ∈ R we have from the given condition
h(x ) = x −1 F (x )x in RC . For any x , y = 0 ∈ R we get from (
1
)
F (x )y + F (y)x = x y−1 F (y)y + yx −1 F (x )x .
(
12
)
F (uy)y + F (y)uy = u F (y)y + u−1 F (uy)uy for all u, y(= 0) ∈ R,
and this implies that
u F (u y)y + u F (y)u y = u2 F (y)y + F (u y)u y for all 0 = u, y ∈ R.
u F (u y) + u F (y)u = u2 F (y) + F (u y)u for all u, y ∈ R.
Since F (x ) = ax + d (x ) for a ∈ Qs ( R), we get from the last relation
uau y + ud (u y) + uayu + ud (y)u = u2ay + u2d (y) + au yu + d (u y)u.
The above relation for y = 1 gives us uau + ud (u) + uau = u2a + au2 + d (u)u. This implies that
[u, d (u)] = [u, [u, a]], hence d (u) = [u, a]; now we get F (x ) = ax + x a − ax = x a and by our assumption
that F (x )x = x h(x ), we get h(x ) = ax , which completes the proof.
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