Abstract
UDC 512.5
Let
R
be a prime ring whose characteristic is not equal to
2
,
let
U
be the Utumi quotient ring of
R
,
and let
C
be the extended centroid of
R
.
Also let
G
and
H
be two generalized derivations on
R
and let
f
(
x
1
,
…
,
x
n
)
be a noncentral multilinear polynomial over
C
.
If
G
(
H
(
u
2
)
)
=
(
H
(
u
)
)
2
for all
u
=
f
(
r
1
,
…
,
r
n
)
,
r
1
,
…
,
r
n
∈
R
,
then one of the following holds:
1)
H
=
0
;
2) there exists
λ
∈
C
such that
G
(
x
)
=
H
(
x
)
=
λ
x
for all
x
∈
R
;
3) there exist
λ
∈
C
and
a
∈
U
such that
H
(
x
)
=
λ
x
and
G
(
x
)
=
[
a
,
x
]
+
λ
x
for all
x
∈
R
and
f
(
x
1
,
…
,
x
n
)
2
is central-valued on
R
.
Publisher
Institute of Mathematics National Academy of Sciences of Ukraine
Subject
General Earth and Planetary Sciences,General Engineering,General Environmental Science