In this paper we prove the following result. Let m 1, n 1 be integers and let R be a 2mn(m+n-1)!-torsion free semiprime ring. Suppose there exist derivations D, G : R R such that D(xm)xn + xnG(xm) = ...0 holds for all x R. In this case both derivations D and G map R into its center and D = -G. We apply this purely algebraic result to obtain a range inclusion result of continuous derivations on Banach algebras.
Let
A
and
B
be Banach algebras with
σ
(
B
)
≠
∅
. Let
θ
,
ϕ
,
γ
∈
σ
(
B
)
and
Der
(
A
×
θ
ϕ
,
γ
B
)
be the set of all linear mappings
d
:
A
×
B
→
A
×
B
satisfying
d
(
(
a
,
b
)
·
θ
(
x
,
y
)
)
=
d
(
...a
,
b
)
·
ϕ
(
x
,
y
)
+
(
a
,
b
)
·
γ
d
(
x
,
y
)
for all
a
,
x
∈
A
and
b
,
y
∈
B
. In this paper, we characterize elements of
Der
(
A
×
θ
ϕ
,
γ
B
)
in the case where
A
has a right identity. We then investigate the concept of centralizing for elements of
Der
(
A
×
θ
ϕ
,
γ
B
)
and determine dependent elements of
Der
(
A
×
θ
ϕ
,
γ
B
)
. We also apply some results to group algebras.
On an additively inverse
MA
-semiring
S
we consider the additive mapping
f
:
S
→
S
satisfying the identity
f
(
x
y
)
=
x
,
f
(
y
)
, where
a
,
b
is a commutator of
a
and
b
. We investigate the ...properties of such a mapping and determine the commutativity of
S
using this mapping.
Let
G
be a locally compact abelian group,
ω
be a weighted function on
R
+
, and let
D
be the Banach algebra
L
0
∞
(
G
)
∗
or
L
0
∞
(
ω
)
∗
. In this paper, we investigate generalized derivations on ...the noncommutative Banach algebra
D
. We characterize
k
-(skew) centralizing generalized derivations of
D
and show that the zero map is the only
k
-skew commuting generalized derivation of
D
. We also investigate the Singer–Wermer conjecture for generalized derivations of
D
and prove that the Singer–Wermer conjecture holds for a generalized derivation of
D
if and only if it is a derivation; or equivalently, it is nilpotent. Finally, we investigate the orthogonality of generalized derivations of
L
0
∞
(
ω
)
∗
and give several necessary and sufficient conditions for orthogonal generalized derivations of
L
0
∞
(
ω
)
∗
.
Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that $u^2{\in}U$ for all $u{\in}U$ and $d$ be a nonzero (${\sigma}$, ${\tau}$)-derivation of ...R. We prove the following results: (i) If $d(u),u_{{\sigma},{\tau}}$ = 0 or $d(u),u_{{\sigma},{\tau}}{\in}C_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$. (ii) If $a{\in}R$ and $d(u),a_{{\sigma},{\tau}}$ = 0 for all $u{\in}U$, then $U{\subseteq}Z$ or $a{\in}Z$. (iii) If $d(u,v)={\pm}u,v_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$. KCI Citation Count: 0
In this paper, we extend some well known results concerning generalized derivations of prime rings to a generalized (σ, τ)-derivation. KCI Citation Count: 0
Let R be a prime ring with characteristic different from two, U a nonzero Lie ideal of R and f be a
generalized derivation associated with d. We prove the following results: (i) If u, f(u) ...∈ Z, for all u ∈ U,
then U ⊂ Z. (ii) (f,d)and(g, h) be two generalized derivations of R such that f(u)v = ug(v), for all
u, v ∈ U, then U ⊂ Z. (iii) f(u, v) = ±u, v, for all u, v ∈ U, then U ⊂ Z.
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