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1. CMB Online first
Generalized Jordan Semiderivations in Prime Rings Let $R$ be a ring, $g$ an endomorphism of $R$.
The additive mapping $d\colon R\rightarrow R$ is called Jordan semiderivation of $R$, associated with $g$, if
$$d(x^2)=d(x)x+g(x)d(x)=d(x)g(x)+xd(x)\quad \text{and}\quad d(g(x))=g(d(x))$$
for all $x\in R$.
The additive mapping $F\colon R\rightarrow R$ is called generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if
$$F(x^2)=F(x)x+g(x)d(x)=F(x)g(x)+xd(x)\quad \text{and}\quad F(g(x))=g(F(x))$$
for all $x\in R$.
In the present paper we prove that
if $R$ is a prime ring of characteristic different from $2$, $g$ an endomorphism of $R$, $d$ a Jordan semiderivation associated with $g$, $F$ a generalized Jordan semiderivation associated with $d$ and $g$,
then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R.$ Moreover, if $R$ is commutative then $F=d$.
Keywords:semiderivation, generalized semiderivation, Jordan semiderivation, prime ring Category:16W25 |
2. CMB 2012 (vol 57 pp. 51)
Jordan $*$-Derivations of Finite-Dimensional Semiprime Algebras In the paper, we characterize Jordan $*$-derivations of a $2$-torsion
free, finite-dimensional semiprime algebra $R$ with involution $*$. To
be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan
$*$-derivation. Then there exists a $*$-algebra decomposition
$R=Uoplus V$ such that both $U$ and $V$ are invariant under
$delta$. Moreover, $*$ is the identity map of $U$ and $delta,|_U$ is a
derivation, and the Jordan $*$-derivation $delta,|_V$ is inner.
We also prove the theorem: Let $R$ be a noncommutative, centrally
closed prime algebra with involution $*$, $operatorname{char},R
e 2$,
and let $delta$ be a nonzero Jordan $*$-derivation of $R$. If $delta$ is
an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and
$delta$ is inner.
Keywords:semiprime algebra, involution, (inner) Jordan $*$-derivation, elementary operator Categories:16W10, 16N60, 16W25 |
3. CMB 2011 (vol 55 pp. 579)
Casimir Operators and Nilpotent Radicals It is shown that a Lie algebra having a nilpotent radical has a
fundamental set of invariants consisting of Casimir operators. A
different proof is given in the well known special case of an
abelian radical. A result relating the number of invariants to the
dimension of the Cartan subalgebra is also established.
Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants Categories:16W25, 17B45, 16S30 |
4. CMB 1999 (vol 42 pp. 401)
Lie Derivations in Prime Rings With Involution Let $R$ be a non-GPI prime ring with involution and characteristic
$\neq 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote
the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$
into itself. Then $\delta=\rho+\epsilon$ where $\epsilon$ is an
additive map into the skew elements of the extended centroid of $R$
which is zero on $[K,K]$, and $\rho$ can be extended to an ordinary
derivation of $\langle K\rangle$ into $RC$, the central closure.
Categories:16W10, 16N60, 16W25 |