In this paper we prove the following result. Let
be a
!-torsion free semiprime ring and let
:
→
be an additive mapping satisfying the relation
+
) = 0 for all
∈
. In this case
= 0.
The aim of this paper is to prove the next result. Let n > 1 be an integer and let R be a n!-torsion free semiprime ring. Suppose that f : R → R is an additive mapping satisfying the relation f(x), ...xn = 0 for all x ∈ R. Then f is commuting on R. KCI Citation Count: 6
The main purpose of this paper is to prove the following result: Let
n
> 1 be a fixed integer, let
R
be a
n
!-torsion free semiprime ring, and let
f
:
R
→
R
be an additive mapping satisfying the ...relation
f
(
x
)
,
x
n
=
…
f
(
x
)
,
x
,
x
,
…
,
x
=
0
for all
x
∈
R
. In this case
f
(
x
),
x
= 0 is fulfilled for all
x
∈
R
. Since any semisimple Banach algebra (for example,
C
∗
algebra) is semiprime, this purely algebraic result might be of some interest from functional analysis point of view.
In this paper we introduce the notion of symmetric skew 3-derivations of prime or semiprime rings and prove that under certain conditions a prime ring with a nonzero symmetric skew 3-derivation has ...to be commutative.
It is well known that there are no nonzero linear derivations on complex commutative semisimple Banach algebras. In this paper we prove the following extension of this result. Let
A
be a complex ...semisimple Banach algebra and let
D
:
A
→
A
be a linear mapping satisfying the relation
D
(
x
2
) = 2
xD
(
x
) for all
x
∈
R
. In this case
D
= 0.
Throughout,
R
will represent an associative ring with center
Z
(
R
). A ring
R
is
n
-torsion free, where
n
> 1 is an integer, if
nx
= 0,
x
∈
R
implies
x
= 0. As usual the commutator
xy
−
yx
will be denoted by
x
,
y
. We shall use the commutator identities
xy
,
z
=
x
,
z
y
+
x
y
,
z
and
x
,
yz
=
x
,
y
z
+
y
x
,
z
for all
x
,
y
,
z
∈
R
. Recall that a ring
R
is prime if for
a
,
b
∈
R
,
aRb
= (0) implies that either
a
= 0 or
b
= 0, and is semiprime in case
aRa
= (0) implies that
a
= 0. An additive mapping
D
is called a derivation if
D
(
xy
) =
D
(
x
)
y
+
xD
(
y
) holds for all pairs
x
,
y
∈
R
, and is called a Jordan derivation in case
D
(
x
2
) =
D
(
x
)
x
+
xD
(
x
) is fulfilled for all
x
∈
R
. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein (8) has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation (see also 1). Cusack (5) has generalized Herstein’s result to 2-torsion free semiprime rings (see 2 for an alternative proof). An additive mapping
D
:
R
→
R
is called a left derivation if
D
(
xy
) =
yD
(
x
) +
xD
(
y
) holds for all pairs
x
,
y
∈
R
and is called a left Jordan derivation (or Jordan left derivation) in case
D
(
x
2
) = 2
xD
(
x
) is fulfilled for all
x
∈
R
. In this paper by a Banach algebra we mean a Banach algebra over the complex field.
In this paper we prove the following result. Let
m
≥ 1,
n
≥ 1 be fixed integers and let
R
be a prime ring with
m
+
n
+ 1 ≤
char
(
R
) or
char
(
R
) = 0. Suppose there exists an additive nonzero ...mapping
D
:
R
→
R
satisfying the relation 2
D
(
x
n
+
m
+1
) = (
m
+
n
+ 1)(
x
m
D
(
x
)
x
n
+
x
n
D
(
x
)
x
m
) for all
. In this case
R
is commutative and
D
is a derivation.
This paper obtains Posner's theorem for products of derivations on left ideals in semiprime rings, so describes when a product of derivations
D and
E of a semiprime ring
R can act as a derivation on ...a left ideal
L of
R. This result yields a quick argument for a theorem of H. Bell and W.S. Martindale on centralizing derivations of
L. Finally results in the literature extending the notion of a centralizing derivation to the situation when
D(
x)
x−
xE(
x) is central for all
x∈
L are obtained as special cases of a more general result here.
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