Let k be a positive integer and R a ring having unit 1. Denote by Z(R) the center of R. Assume that the characteristic of R is not 2 and there is an idempotent element e∈R such that R satisfies ...aRe={0}⇒a=0, aR(1−e)={0}⇒a=0, Z(eRe)k=Z(eRe) and Z((1−e)R(1−e))k=Z((1−e)R(1−e)). Then every additive map f:R→R is k-commuting if and only if f(x)=αx+h(x) for all x∈R, where α∈Z(R) and h is an additive map from R into Z(R). As applications, all k-commuting additive maps on prime rings and von Neumann algebras are characterized.
Let
be a natural number. Let
be the ring of all
matrices over an arbitrary field
. In the present paper, we will find the description of all additive mappings
such that
for all
, where
(
and
are ...matrices units). As an application we will characterize all additive maps
such that
for all
, where
represents the set formed by all rank-1 matrices of
. Precisely, we will show that
for all
if and only if
for all
.
Let SVir be the well-known super-Virasoro algebras. In this paper, we first prove that any super-skewsymmetric super-biderivation of SVir is inner. Based on this, we show that every linear ...super-commuting map ψ on SVir is of the form ψ(x) = f(x)c, where f is a linear function from SVir to ℂ mapping the odd part of SVir to zero, and c is the central charge of SVir.
In this article, we compute biderivations of the q-deformed Heisenberg-Virasoro algebra. As an application, linear commuting maps are characterized. We also provide calculations of α-derivations and ...α-biderivations of the q-deformed Heisenberg-Virasoro algebra and obtain that it has no nonzero α-derivations and α-biderivations.
The n-th Schrödinger algebra
is the semi-direct of the Lie algebra
with the n-th Heisenberg Lie algebra
. In this paper, all derivations and biderivations of the n-th Schrödinger algebra are ...determined. As applications, all linear commuting maps and commutative post-Lie algebra structures on
are obtained.
Let m, n be integers such that 1<m<n. Let
$ \mathcal {R}=M_n(\mathbb {D}) $
R
=
M
n
(
D
)
be the ring of all
$ n\times n $
n
×
n
matrices over a division ring
$ \mathbb {D} $
D
,
$ \mathcal {M} $
M
...an additive subgroup of
$ \mathcal {R} $
R
and
$ G:\mathcal {R}^m\rightarrow \mathcal {R} $
G
:
R
m
→
R
an m-additive map. In this paper, under a mild technical assumption, we prove that
$ \delta _1(x)=G(x,\ldots,x)\in \mathcal {M} $
δ
1
(
x
)
=
G
(
x
,
...
,
x
)
∈
M
for each rank-s matrix
$ x\in \mathcal {R} $
x
∈
R
implies
$ \delta _1(x)\in \mathcal {M} $
δ
1
(
x
)
∈
M
for each
$ x\in \mathcal {R} $
x
∈
R
, where s is a fixed integer such that
$ m\leq s \lt n $
m
≤
s
<
n
, which has been considered for the case s = n in Xu X, Zhu J., Central traces of multiadditive maps on invertible matrices, Linear Multilinear Algebra 2018; 66:1442-1448. Also, an example is provided showing that the conclusion will not be true if s<m. As applications, we also extend the conclusions by Liu, Franca et al., Lee et al. and Beidar et al., respectively, to the case of rank-s matrices for
$ m\leq s \lt n $
m
≤
s
<
n
.
Let Nr, r≥4, be the ring of strictly upper triangular matrices with entries in a field F of characteristic zero. We describe all linear maps f:Nr→Nr satisfying f(x),x=0 for every x∈Nr.
Suppose the ground field is algebraically closed and of characteristic different from 2. In this paper, we described the intrinsic connections among linear super-commuting maps, super-biderivations ...and centroids for Lie superalgebras satisfying certain assumptions. This is a generalization of the results of Bresar and Zhao on Lie algebras.
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