In this paper we prove the following result: let $m,n\geq 1$ be distinct integers, let $R$ be an $mn(m+n)|m-n|$-torsion free semiprime ring and let $D:R\rightarrow R$ be an $(m,n)$-Jordan derivation, ...that is an additive mapping satisfying the relation $(m+n)D(x^{2})=2mD(x)x+2nxD(x)$ for $x\in R$. Then $D$ is a derivation which maps $R$ into its centre.
For prime algebras, we describe a linear map which behaves like a left derivation on a fixed multilinear polynomial in noncommuting indeterminates and, in particular, we characterize left derivations ...by their action on mth powers.
The purpose of this paper is to prove the following result. Let m, n ≥ 1 be some fixed integers with m ≠ n, and let R be a prime ring with (m + n)² < char (R). Suppose a nonzero additive mapping D : ...R → R exists satisfying the relation (m + n)² D(x³) = m(3m + n) D(x)x² + 4mnxD(x)x + n(3n + m)x² D(x) for all x ϵ R. In this case D is a derivation and R is 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.
In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let δ be a spectrally bounded left Jordan derivation on a Banach algebra А. Then δ maps Аinto its ...Jacobson radical. (ii) Let δ be a left Jordan derivation on a unital Banach algebra А with the condition sup$\{r(c-1δ(c)): c ∈ A invertible} < ∞. Then δ maps А into its Jacobson radical.
Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in \citep.\,260{Ash08}: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation. KCI Citation Count: 2
The purpose of this paper is to prove the following result. Let
≥ 1,
≥ 1 be some fixed integers with
≠
and let
be a prime ring with
) ≠ 2
) |
−
| . Suppose there exists a nonzero additive mapping
:
→
...satisfying the relation (
+
) = 2
+ 2
) for all
∈
((
)-Jordan derivation). If either
) = 0 or
) > 3 then
is a derivation and
is commutative.
PDF
