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
Let
R be a maximal order and
A,
B be
R-ideals of
R. Clearly
(
A
B
)
∗
⊇
B
∗
A
∗
is satisfied and if
R is a Dedekind prime ring, the equality holds, i.e.,
(
A
B
)
∗
=
B
∗
A
∗
. However, the equality ...is not true in general. In this paper, we answer the question: If
R is a maximal order when is
(
A
B
)
∗
=
B
∗
A
∗
for all non-zero
R-ideals of
R? We call prime Noetherian maximal orders satisfying this property,
generalized Dedekind prime rings. We give several characterizations of G-Dedekind prime rings and show that being a G-Dedekind prime ring is a Morita invariant. Moreover, we prove that if
R is a PI G-Dedekind prime ring then the polynomial ring
R
x
and the Rees ring
R
X
t
associated to an invertible ideal
X are also PI G-Dedekind prime rings.
Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X and let A(X)⊆L(X) be a standard operator algebra. Suppose there exists a linear mapping ...T:A(X)→L(X) satisfying the relation T(An)=T(A)An−1−AT(An−2)A−An−1T(A) for all A∈A(X), where n>2 is some fixed integer. Then T is of the form: (i)T(A)=0 for all A∈F(X) and (ii) T(A)=BA, for all A∈A(X) and some B∈L(X).
Let
R
be a prime ring of characteristic different from
2
,
U
its right Utumi quotient ring,
C
its extended centroid,
G
a non-zero generalized derivation of
R
,
a
≠
0
be an element of
R
,
I
a non-zero ...right ideal of
R
such that
s
4
(
I
,
…
,
I
)
I
≠
0
and
n
,
k
≥
1
fixed integers. If
a
G
(
r
1
,
r
2
n
)
,
r
1
,
r
2
n
k
=
0
, for any
r
1
,
r
2
∈
I
, then either there exist
c
∈
U
and
γ
∈
C
, such that
G
(
x
)
=
c
x
and
(
c
-
γ
)
I
=
0
, or
a
I
=
a
G
(
I
)
=
0
.
Let
R
be a prime ring with its Utumi ring of quotients
U
,
F
a nonzero generalized derivation of
R
and
L
a noncentral Lie ideal of
R
. Suppose that
F
(
u
n
1
)
,
u
n
2
,
u
n
3
,
…
,
u
n
k
=
0
for ...all
u
∈
L
, where
n
1
,
n
2
,
…
,
n
k
≥
1
are fixed integers. Then one of the following holds:
there exists
α
∈
C
such that
F
(
x
)
=
α
x
for all
x
∈
R
;
R
satisfies
s
4
, the standard identity in four variables.
Also we study the situation, when
x
∈
I
,
I
, where
I
is a nonzero left ideal of
R
. As an application we obtain some range inclusion results of continuous generalized derivations on Banach algebras.
Let R be an associative ring. In the present paper, we investigate commutativity of a ring admitting an additive mapping F satisfying any one of the following properties: (i) F (x, y
2
) = F (x
2
, y
...2
), (ii) F ((x ◦ y)
2
) = F (x
2
◦ y
2
), (iii) F ((xy)
n
) = F (x
n
y
n
), (iv) F (x
m
y
n
) = F (y
n
x
m
), (v) (F (x)F (y))
n
= (F (y)F (x))
n
for all x, y ∈ R, where m and n are positive integers greater than 1. Moreover, some related results are also discussed. Finally, some examples are given to demonstrate that the restrictions imposed on the hypotheses of the various results are not superfluous.
Let
R
be a prime ring of characteristic different from 2 and 3,
Q
r
its right Martindale quotient ring,
C
its extended centroid,
L
a non-central Lie ideal of
R
and
n
≥ 1 a fixed positive integer. Let
...α
be an automorphism of the ring
R
. An additive map
D
:
R
→
R
is called an
α
-derivation (or a skew derivation) on
R
if
D
(
xy
) =
D
(
x
)
y
+
α
(
x
)
D
(
y
) for all
x
,
y
∈
R
. An additive mapping
F
:
R
→
R
is called a generalized
α
-derivation (or a generalized skew derivation) on
R
if there exists a skew derivation
D
on
R
such that
F
(
xy
) =
F
(
x
)
y
+
α
(
x
)
D
(
y
) for all
x
,
y
∈
R
.
We prove that, if
F
is a nonzero generalized skew derivation of
R
such that
F
(
x
)×
F
(
x
),
x
n
= 0 for any
x
∈
L
, then either there exists λ ∈
C
such that
F
(
x
) = λ
x
for all
x
∈
R
, or
R
⊆
M
2
(
C
) and there exist
a
∈
Q
r
and λ ∈
C
such that
F
(
x
) =
ax
+
xa
+ λ
x
for any
x
∈
R
.
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