On generalized derivations satisfying certain identities
Let $R$ be a prime ring with char $R \neq 2$ and $d$ be a generalized derivation on $R$. The goal of this study is to investigate the generalized derivation $d$ satisfying any one of the following identities: $$(i) \quad d[(x, y)] = [d(x), d(y)] \quad \text{for all} x, y \in R;$$ $$(ii) \quad d[(x,...
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2011
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| author | Albaş, E. Албас, Е. |
| author_facet | Albaş, E. Албас, Е. |
| author_sort | Albaş, E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:35:13Z |
| description | Let $R$ be a prime ring with char $R \neq 2$ and $d$ be a generalized derivation on $R$. The goal of this study
is to investigate the generalized derivation $d$ satisfying any one of the following identities:
$$(i) \quad d[(x, y)] = [d(x), d(y)] \quad \text{for all} x, y \in R;$$
$$(ii) \quad d[(x, y)] = [d(y), d(x)] \quad \text{for all} x, y \in R;$$
$$(iii)\quad d([x, y]) = [d(x), d(y)] \text{either} d([x, y]) = [d(y), d(x)] \quad \text{for all} x, y \in R$$. |
| first_indexed | 2026-03-24T02:29:32Z |
| format | Article |
| fulltext |
© E. ALBAŞ, 2011
596 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
UDC 512.5
E. Albaş (Ege Univ., Izmir, Turkey)
ON GENERALIZED DERIVATIONS SATISFYING
CERTAIN IDENTITIES
ПРО УЗАГАЛЬНЕНІ ДИФЕРЕНЦІЮВАННЯ,
ЩО ЗАДОВОЛЬНЯЮТЬ ДЕЯКІ ТОТОЖНОСТІ
Let R be a prime ring with char R ! 2 and d be a generalized derivation on R . The goal of this study
is to investigate the generalized derivation d satisfying any one of the following identities:
(i) d[(x, y)] = [d(x), d(y)] for all x , y ! R ;
(ii) d[(x, y)] = [d(y), d(x)] for all x , y ! R ;
(iii) either d([x, y]) = [d(x), d(y)] or d([x, y]) = [d(y), d(x)] for all x , y ! R .
Припустимо, що R — просте кільце з R ! 2 , а d — узагальнене диференціювання на R . Мета
цієї роботи полягає у дослідженні диференціювання d , що задовольняє будь-яку з наступних тотож-
ностей:
(i) d[(x, y)] = [d(x), d(y)] для всіх x , y ! R ;
(ii) d[(x, y)] = [d(y), d(x)] для всіх x , y ! R ;
(iii) d([x, y]) = [d(x), d(y)] або d([x, y]) = [d(y), d(x)] для всіх x , y ! R .
1. Introduction. Let R always denote an associative ring with center Z , extended
centroid C , Utumi quotient ring U . Recall that an additive mapping ! : R! R is
called a derivation if !(xy) = !(x)y + x!(y) holds for all x , y !R . The study of
prime rings with derivations was initiated by Posner [16]. Many related generalizations
have been done on this subject (see [16, 8], where further references can be found).
Following Bresar [8], d : R! R is called a generalized derivation if there exists a
derivation ! of R such that d(xy) = d(x)y + x!(y) for all x , y !R . It is clear
that the concept of generalized derivations covers both the concepts of a derivation and
of a left multiplier (i.e., an additive mapping f : R! R satisfying f (xy) = f (x)y
for all x , y !R ). In [10], Hvala initiated the study of generalized derivations from the
algebraic viewpoint. Many authors have studied generalized derivations in the context
of prime and semiprime rings (see [1– 4, 13, 14, 17]). In [13], T. K. Lee extended the
definition of generalized derivations as follows: By a generalized derivation we mean
an additive mapping d : I !U such that d(xy) = d(x)y + x!(y) for all x , y ! I ,
where U is the right Utumi quotient ring, I is a dense right ideal of R and ! is a
derivation from I into U . Moreover Lee also proved that every generalized deriva-
tion can be uniquely extended to a generalized derivation of U and thus all general-
ized derivations of R will be implicitly assumed to be defined on the whole U and
he obtained the following results:
Theorem ([13], Theorem 3). Every generalized derivation d on a dense right
ideal of R can be uniquely extended to U and assumes the form d(x) = ax + !(x)
for some a !U and a derivation ! on U .
Over the last three decades, several authors have proved the commutativity theo-
rems for prime or semiprime rings admitting derivations or generalized derivations sat-
ON GENERALIZED DERIVATIONS SATISFYING CERTAIN IDENTITIES 597
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
isfying some relations (see [3, 4, 7, 17]). In [4], M. Ashraf et al. investigated the com-
mutativity of a prime ring R admitting a generalized derivation F with associated
derivation d satisfying any one of the following conditions: d(x) ! F(y) = 0 , [d(x),
F(y)] = 0 , d(x) ! F(y) = x ! y , d(x)! F(y) + x ! y = 0 , d(x)! F(y) ! xy "Z ,
d(x)! F(y) + xy !Z , [d(x), F(y)] = [x, y] , d(x), F(y)[ ] + [x, y] = 0 for all x ,
y ! I , where I is a nonzero ideal of R , [x, y] = xy ! yx and x ! y = xy + yx . In
[3], the authors proved the commutativity of a prime ring R in which a generalized
derivation F satisfies any one of the following properties: (i) F(xy) ! xy "Z ,
(ii) F(xy) + xy !Z , (iii) F(xy) ! yx "Z , (iv) F(xy) + yx !Z , (v) F(x)F(y) –
– xy !Z and (vi) F(x)F(y) + xy !Z , for all x , y !R . In [17], Shuliang proved that
if L is a lie ideal of a prime ring R such that u2 !L for all u !L and if F is a
generalized derivation on R associated with a derivation d on R satisfying any
one of the following conditions: (1) d(x) ! F(y) = 0 , (2) d(x), F(y)[ ] = 0, (3) either
d(x) ! F(y) = x ! y or d(x) ! F(y) + x ! y = 0 , (4) either d(x)! F(y) = [x, y] or
d(x) ! F(y) + [x, y] = 0 , (5) either d(x) ! F(y) ! xy "Z or d(x) ! F(y) + xy !Z ,
(6) d(x), F(y)[ ] = [x, y] or d(x), F(y)[ ] + [x, y] = 0 , (7) either d(x), F(y)[ ] = x ! y
or d(x), F(y)[ ] + x ! y = 0 for all x , y !L , then either d = 0 or L ! Z .
In this paper we aim to investigate the generalized derivation d on a prime ring
R associated with a derivation ! on satisfying any one of the following identities:
(i) d [x, y]( ) = d(x), d(y)[ ] for all x , y !R , (ii) d [x, y]( ) = d(y), d(x)[ ] for all x ,
y !R , (iii) either d [x, y]( ) = d(x), d(y)[ ] or d [x, y]( ) = d(y), d(x)[ ] for all x ,
y !R .
In all that follows, unless stated otherwise, R will be a prime ring. The related ob-
ject we need to mention is the two-sided Quotient ring Q of R , the right Utumi quo-
tient ring U of R (sometimes, as in [6], U is called the maximal ring of quotients).
The definitions, the axiomatic formulations and the properties of this quotient ring U
can be found in [6] and [5].
We make a frequent use of the theory of generalized polynomial identities and dif-
ferential identities (see [6, 9, 11, 12, 15]). In particular we need to recall that when R
is a prime ring and I a nonzero two-sided ideal of R , then I , R , Q and U
satisfy the same generalized polynomial identities [9] and also the same differential
identities [12].
We will also make frequent use of the following result due to Kharchenko [11] (see
also [12]):
Let R be a prime ring, d a nonzero derivation of R and I a nonzero two-
sided ideal of R . Let f x1,…, xn , d(x1),…, d(xn )( ) be a differential identity in I ,
that is the relation
f r1,…, rn , d(r1),…, d(rn )( ) = 0
holds for all r1,…, rn ! I . Then one of the following holds:
598 E. ALBAŞ
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
1) either d is an inner derivation in Q , the Martindale quotient ring of ��� R , in
the sense that there exists q !Q such that d(x) = [q, x] , for all x !R , ���and I
satisfies the generalized polynomial identity
f r1,…, rn , [q, r1],…, [q, rn ]( ) ;
2) or I satisfies the generalized polynomial identity
f x1,…, xn , y1,…, yn( ) .
In [14], T. K. Lee and W. K. Shiue proved a version of Kharchenko’s theorem for
generalized derivations and presented some results concerning certain identities with
generalized derivations. More details about generalized derivations can be found in [10,
11, 13, 14].
2. The results. ln the following, we assume that R is a prime ring with char
R ! 2 and that Z is the center of R unless stated otherwise. We denote the iden-
tity map of a ring R by Iid (i.e., the map Iid : R! R defined by Iid (x) = x for
all x !R ). By a map !Iid : R" R we mean the map defined by (!Iid )(x) = !x for
all x !R .
We begin with the following.
Lemma 1. Let R be a prime ring with char R ! 2 and d be a generalized
derivation on R associated with a derivation ! on R . If d [x, y]( ) = d(x), d(y)[ ]
holds for all x , y !R then either R is commutative, or d = 0 , or d = Iid .
Proof. As we stated as theorem we can take the generalized derivation d as the
form d(x) = ax + !(x) where a !U and ! is a derivation on U .
If ! = 0 , then by the hypothesis we have a[x, y] = [ax, ay] for all x , y !R .
Replacing yz by y we have ay[x, z] = ay[ax, z] , hence ay[x ! ax, z] = 0 for all
x , y , z !R . By the primeness of R we get either a = 0 or [x ! ax, z] = 0 for all
x, z !R . The first case gives us that d = 0 , as desired. For the second case, let
[x ! ax, z] = 0 for all x, z !R . Substituting xyr by x we have 0 =
= (xy ! axy)r, z[ ] = (xy ! axy)[r, z] = (x ! ax)y[r, z] for all x , y , r !R . By the
primeness of R we obtain that either R is commutative, or x ! ax = 0 for all
x !R implying that d(x) = ax = x , i.e., d = Iid , as desired.
Now we may consider the case that R is not commutative. Suppose ! " 0 .
Since R and U satisfy the same differential identities [12], we get
a[x, y] + !(x), y[ ] + x,!(y)[ ] = ax + !(x), ay + !(y)[ ] for all x , y !U . (1)
In light of Kharchenko’s theory [11] we can divide the proof into two cases.
Assume first that ! is an outer derivation of U . By Kharchenko’s theorem in
[11, l2], we get
a[x, y] + [z, y] + [x,w] = [ax, ay] + [ax,w] + [z, ay] + [z,w]
for all x , y , z , w !U . In particular, taking w = z = 0 we obtain a[x, y] =
= [ax, ay] . By the same argument as above we have either R is commutative or
ON GENERALIZED DERIVATIONS SATISFYING CERTAIN IDENTITIES 599
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
a = 0 . Let a = 0 . Using this fact and taking w = y in the above relation we have
[x,w] = 0 for all x,w !U implying R is commutative. It is seen that the two
cases give us a contradiction.
Assume now that a is an inner derivation of U induced by an element q !U , that
is !(x) = [q, x] , for all x !U . In this case d(x) = ax + !(x) = ax + [q, x] . Re-
placing 1 for y in (1) we have
ax + !(x), a[ ] = 0 for all x !U . (2)
Replacing q by x in (2) we get [aq, a] = a[q, a] = 0 , i.e., a!(a) = 0 . Using (2),
we have
a[x, a] + !(x), a[ ] = 0 for all x !U . (3)
Taking rx in place of x in (3)
ar[x, a] + !(r)[x, a] + [r, a]!(x) + r !(x), a[ ] = 0 for all x , r !U . (4)
Say !(x) = [a, x] , x !U . By (3) we have 0 = a[x, a] + !(x)a " a!(x) =
= !a "(x) + #(x)( ) + #(x)a for all x !U . Hence we get
a !(x) + "(x)( ) = "(x)a for all x !U . (5)
By (3) we have r !(x), a[ ] = "ra[x, a] for all r , x !U . Using this fact in (4) we
arrive at 0 = ar[x, a] + !(r)[x, a] + [r, a]!(x) " ra[x, a] = [a, r][x, a] + !(r)[x, a] +
+ [r, a]!(x) = !"(r)"(x) ! #(r)"(x) ! "(r)#(x) . The last relation implies that
!(r) + "(r)( )!(x) + !(r)"(x) = 0 for all r , x !U . (6)
Multiplying (6) by a from the left-hand side and using (5) we find that 0 = a(!(r) +
+ !(r))"(x) + a!(r)"(x) = !(r)a"(x) + a"(r)!(x) , i.e.,
!(r)a"(x) + a"(r)!(x) = 0 for all r , x !U . (7)
Substituting zx by x in (7) and using (7) we have
!(r)az"(x) + a"(r)z!(x) = 0 .
Taking !(z) instead of z in the last relation and using (7) again we get
!(r)a !(z)"(x) # "(z)!(x)( ) = 0 .
Replacing rs by r we arrive at
!(r)sa !(z)"(x) # "(z)!(x)( ) = 0 .
Since U is prime and ! " 0 we obtain a !(z)"(x) # "(z)!(x)( ) = a!(z)"(x) –
– a!(z)"(x) = 0 . Using (7) in the last relation we have a!(z) + !(z)a( )"(x) = 0 for
all x, z !U . Substituting rx by x in the last relation we get
600 E. ALBAŞ
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
a!(z) + !(z)a( ) r"(x) = 0 for all x, z !U .
By the primeness of U we obtain that either ! = 0 , or a!(z) + !(z)a = 0 for all
z !U .
The first case implies that a !C . Using this fact in (1) we have
(a ! a2 )[x, y] + (1! a) ["(x), y] + [x,"(y)]( ) = "(x),"(y)[ ] for all x , y !U .
(8)
Replacing q by y in (8) and using the facts that !(x) = [q, x] and !2 (x) =
= q,!(x)[ ] we get
(a ! a2 )"(x) + (1 ! a)"2 (x) = 0 .
Taking xy for x and using a !C we have 2(1 ! a)"(x)"(y) = 0 . Since char
R ! 2 and a !C we have either !(x)!(y) = 0 for all x , y !U or a = 1 . If
!(x)!(y) = 0 , then taking ry for y we get !(x)r!(y) = 0 implying that ! = 0
by the primeness of U , a contradiction. If a = 1 , then we find !(x),!(y)[ ] = 0 .
Substituting yq by y in the last relation we have !(y)!2 (x) = 0 for all x ,
y !U . Since ! " 0 and U is prime we get !2 (x) = 0 , implying that ! = 0 , a
contradiction.
So we are forced to conclude that
a!(z) + !(z)a = 0 for all z !U . (9)
Using (9) in (3) we have 0 = a[x, a] + !(x)a " a!(x) = !a"(x) ! a#(x) ! a#(x) =
= !a "(x) + 2#(x)( ) . Hence we get a !(x) + 2"(x)( ) = 0 . Replacing rx by x in
the last relation and using the primeness of U we obtain that either a = 0 or
!(x) = "2#(x) for all x !U .
If a = 0 , (1) is reduced to
!(x), y[ ] + x,!(y)[ ] = !(x),!(y)[ ] .
Substituting q by y we have !2 (x) = 0 , implying that ! = 0 , a contradiction.
So we arrive at the case !(x) = "2#(x) for all x !U . Replacing yx by y in the
hypothesis we get
[x, y]!(x) = d(y) d(x), x[ ] + d(x), y[ ]!(x) + y d(x),!(x)[ ] for all x , y !U .
(10)
Taking yz instead of y in (10) and using (10) we have
x ! d(x), y[ ] z"(x) = [a, y]z d(x), x[ ] + "(y)z d(x), x[ ] .
Since !(x) = [a, x] = "2#(x) and char R ! 2 we get !(a) = 0 . Using this fact and
taking a in place of y in the above relation we obtain that
ON GENERALIZED DERIVATIONS SATISFYING CERTAIN IDENTITIES 601
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
x ! d(x), a[ ] z"(x) = 0 for all x , z !U .
By the primeness of U we have that for each x !U , either [x ! d(x), a] = 0 or
!(x) = 0 . Let H = x !U : x " d(x), a[ ] = 0{ } and K = x !U : "(x) = 0{ } . It is
clear that (H , +) and (K , +) are two additive subgroup of (U, +) such that (U, +) =
= (H , +) ! (K , +) . But a group can not be the union two proper subgroups. There-
fore we get either U = H or U = K . Since ! " 0 we arrive at x ! d(x), a[ ] = 0
for all x !U . By (3) the last relation implies that 0 = [x, a] ! d(x), a[ ] = [x, a] –
– a[x, a] + [!(x), a]( ) = [x, a] = !"(x) . Hence this last relation yields !(x) = 0
whence !(x) = 0 , a contradiction.
Remark 1. If ! is a derivation on a ring R then the map ! " : R# R
���defined by (!")(x) = !"(x) is also a derivation on R . Similarly, if d is a general-
ized derivation on a ring R associated with a derivation ! on R then a ���map
!d : R" R defined by (!d)(x) = !d(x) is also a generalized derivation ��� on R as-
sociated with a derivation !" on R .
Lemma 2. Let R be a prime ring with char R ! 2 and d be a generalized
derivation on R associated with a derivation ! on R . If d [x, y]( ) = d(y), d(x)[ ]
holds for all x , y !R then either R is commutative, or d = 0 , or d = !Iid .
Proof. Let d [x, y]( ) = d(y), d(x)[ ] for all x , y !R . Replace !x by x .
Since
d [!x, y]( ) = d  = !d [x, y]( ) = (!d) [x, y]( )
and
d(y), d(!x)[ ] = d(y), !d(x)[ ] =
= ! d(y), d(x)[ ] = d(x), d(y)[ ] = !d(x), !d(y)[ ] = (!d)x, (!d)y[ ]
we have (!d) [x, y]( ) = (!d)(x), (!d)(y)[ ] for all x, y !R . In view of Remark 1
and Lemma 1 we obtain that either R is commutative, or d = 0 , or d = !Iid .
Theorem 1. Let d be a generalized derivation on R be a prime ring with char
R ! 2 and R associated with a derivation ! on R . If d satisfies either
d [x, y]( ) = d(x), d(y)[ ] or d [x, y]( ) = d(y), d(x)[ ] for all x , y !R then either
R is commutative, or d = 0 , or d = Iid , or d = !Iid .
Proof. For each x !R we set I x = y !R : d([x, y]) = [d(x), d(y)]{ } and Jx =
= y !R : d [x, y]( ) = d(y), d(x)[ ]{ } . It is clear that for each x !R , I x and Jx are
two additive subgroup of R and (R, +) = (I x , +)! (Jx , +) . But a group can not be
the union two proper subgroups. So we are forced to conclude that either R = I x or
R = Jx . Now we set I = {x !R : R = I x} and J = {y !R : R = Jx} . The sets I
and J are also two subgroups of R and (R, +) = (I , +)! (J, +) . By the similar
602 E. ALBAŞ
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
manner as above we have R = I or R = J . By Lemmas 1 and 2 we obtain desired
results.
Example 1. Consider the matrix ring R =
x y
0 0
!
"
#
$
%
& : x, y 'Z
(
)
*
+*
,
-
*
.*
, where Z is
the set of all integers. It is clear to see that a map ! : R" R defined by
!
x y
0 0
"
#
$
%
&
'
(
)
*
+
,
- =
0 x
0 0
!
"
#
$
%
& is a derivation on R . Then a map d : R! R defined by
d
x y
0 0
!
"
#
$
%
&
'
(
)
*
+
, =
x x + y
0 0
!
"
#
$
%
& is a generalized derivation associated with ! satisfy-
ing the condition d [X,Y ]( ) = d(X), d(Y )[ ] for all X , Y !R , but neither R is
commutative, nor d = 0 , nor d = Iid .
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P. 1 – 6.
Received 08.10.10
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| id | umjimathkievua-article-2745 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:29:32Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/09b04ab80e7c6576c6ccd8e40486550d.pdf |
| spelling | umjimathkievua-article-27452020-03-18T19:35:13Z On generalized derivations satisfying certain identities Про узагальнені диференціювання, що задовольняють деякі тотожності Albaş, E. Албас, Е. Let $R$ be a prime ring with char $R \neq 2$ and $d$ be a generalized derivation on $R$. The goal of this study is to investigate the generalized derivation $d$ satisfying any one of the following identities: $$(i) \quad d[(x, y)] = [d(x), d(y)] \quad \text{for all} x, y \in R;$$ $$(ii) \quad d[(x, y)] = [d(y), d(x)] \quad \text{for all} x, y \in R;$$ $$(iii)\quad d([x, y]) = [d(x), d(y)] \text{either} d([x, y]) = [d(y), d(x)] \quad \text{for all} x, y \in R$$. Припустимо, що $R$ — просте кільце з $R \neq 2$, а $d$ — узагальнене диференціювання на $R$. Мета цієї роботи полягає у дослідженні диференціювання $d$, що задовольняє будь-яку з наступних тотожностей: $$(i) d[(x, y)] = [d(x), d(y)] \quad \text{для всіх} x, y \in R;$$ $$(ii) d[(x, y)] = [d(y), d(x)] \quad \text{для всіх} x, y \in R;$$ $$(iii) d([x, y]) = [d(x), d(y)] \text{або} d([x, y]) = [d(y), d(x)] \quad \text{для всіх} x, y \in R$$. Institute of Mathematics, NAS of Ukraine 2011-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2745 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 5 (2011); 596-602 Український математичний журнал; Том 63 № 5 (2011); 596-602 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2745/2246 https://umj.imath.kiev.ua/index.php/umj/article/view/2745/2247 Copyright (c) 2011 Albaş E. |
| spellingShingle | Albaş, E. Албас, Е. On generalized derivations satisfying certain identities |
| title | On generalized derivations satisfying certain identities |
| title_alt | Про узагальнені диференціювання, що задовольняють деякі тотожності |
| title_full | On generalized derivations satisfying certain identities |
| title_fullStr | On generalized derivations satisfying certain identities |
| title_full_unstemmed | On generalized derivations satisfying certain identities |
| title_short | On generalized derivations satisfying certain identities |
| title_sort | on generalized derivations satisfying certain identities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2745 |
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