According to Posner’s second theorem, a prime ring is forced to be commutative if a nonzero centralizing derivation exists on it. In this article, we extend this result to prime rings with ...antiautomorphisms and nonzero skew derivations. Additionally, a case is shown to demonstrate that the restrictions placed on the theorems’ hypothesis were not unnecessary.
Here, we investigate symmetric bi-derivations and their generalizations on L?
0 (G)*. For k ? N, we show that if B:L?0(G)*x L?0(G)* ? L?0(G)* is
asymmetric bi-derivation such that B(m,m),mk ? ...Z(L?0(G)*) for all m ? L?
0 (G)*, then B is the zero map. Furthermore, we characterize symmetric
generalized biderivations on group algebras. We also prove that any
symmetric Jordan bi-derivation on L? 0(G)* is a symmetric bi-derivation.
Let R be a semiprime ring and I a nonzero ideal of R. A map F:R→R is called a multiplicative generalized derivation if there exists a map d:R→R such that F(xy) = F(x)y+xd(y), for all x,y∈R. In the ...present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i)
ii)
iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I.
A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. The main purpose of this paper is to prove a *-version ...of Posner's theorem mentioned above. Moreover, we describe the structure of an arbitrary additive mapping which is *-centralizing on a prime ring with involution.
In this paper, we investigate Jordan derivations, Jordan right derivations and Jordan left derivations of
L
0
∞
(
G
)
∗
. We show that any Jordan (right) derivation on
L
0
∞
(
G
)
∗
is a (right) ...derivation on
L
0
∞
(
G
)
∗
and the zero map is the only Jordan left derivation on
L
0
∞
(
G
)
∗
. Then, we prove that the range of a Jordan (right) derivation on
L
0
∞
(
G
)
∗
is contained into
rad
(
L
0
∞
(
G
)
∗
)
. Finally, we establish that the product of two Jordan (right) derivations of
L
0
∞
(
G
)
∗
is always a derivation on
L
0
∞
(
G
)
∗
and there is no nonzero centralizing Jordan (right) derivation on
L
0
∞
(
G
)
∗
.
Let $R$ be a prime ring (or semiprime ring) with center $Z(R)$, $I$ a nonzero ideal of $R,$ $T$ an automorphism of $R$, $S:R^{n}\rightarrow R$ be a symmetric skew $n$-derivation associated with the ...automorphism $T$ and $ \Delta $ is the trace of $S.$ In this paper, we shall prove that $ S(x_{1},\ldots ,x_{n})=0$ for all $x_{1},\ldots ,x_{n}\in R$ if any one of the following holds: i) $\Delta (x)=0,$ ii) $\Delta (x),T(x)=0$ for all $ x\in I.$ Moreover, we prove that if $\Delta (x),T(x)\in Z(R)$ for all $x\in I,$ then $R$ is a commutative ring. KCI Citation Count: 0
Let
be a prime ring with centre
,
a non-zero Lie ideal of
, and σ a non-trivial automorphism of
such that
for all
. If
, then it is shown that
satisfies
, the standard identity in four variables.
In this paper we prove the following result. Let R be a 2-torsion free semiprime ring and let f : R → R be an additive mapping satisfying the relation f(x)x
+ x
f(x) = 0 for all x ∈ R. In this case f ...= 0. Any semisimple Banach algebra (for example, C* algebra) is semiprime. Therefore this algebraic result might be of some interest from functional analysis point of view.
Let
be a semiprime ring with center
and extended centroid
. For a fixed integer
≥ 2, the trace
of a permuting
-additive mapping
is defined as
for all
∈
.
The notion of permuting
-derivation was ...introduced by Park
J. Chungcheong Math. Soc. 22 (2009), no.3, 451–458 as follows:
a permuting
-additive mapping
is said to be permuting
-derivation if
A permuting
-additive mapping
is known to be
a permuting generalized
-derivation if there exists a permuting
-derivation
such that
The main result of this paper states that if
is a nonzero ideal of a semiprime ring
and
is a permuting
-derivation such that
and
for all
∈
, where δ is the trace of Δ, then
contains a nonzero central ideal. Furthermore, some related results are also proven.
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.
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