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1. 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 |
2. CMB 2011 (vol 56 pp. 344)
Involutions and Anticommutativity in Group Rings Let $g\mapsto g^*$ denote an involution on a
group $G$. For any (commutative, associative) ring
$R$ (with $1$), $*$ extends linearly to an involution
of the group ring $RG$. An element $\alpha\in RG$
is symmetric if $\alpha^*=\alpha$ and
skew-symmetric if $\alpha^*=-\alpha$.
The skew-symmetric elements are closed under
the Lie bracket, $[\alpha,\beta]=\alpha\beta-\beta\alpha$.
In this paper, we investigate when this set is also closed
under the ring product in $RG$.
The symmetric elements are closed under the Jordan
product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$.
Here, we determine when this product is trivial.
These two problems
are analogues of problems about the skew-symmetric and
symmetric elements in group rings that have received a
lot of attention.
Categories:16W10, 16S34 |
3. CMB 2007 (vol 50 pp. 105)
On Valuations, Places and Graded Rings Associated to $*$-Orderings We study natural $*$-valuations, $*$-places and graded $*$-rings
associated with $*$-ordered rings.
We prove that the natural $*$-valuation is always quasi-Ore and is
even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is
commutative), provided the ring contains an imaginary unit.
Furthermore, it is proved that the graded $*$-ring is isomorphic
to a twisted semigroup algebra. Our results are applied to answer a question
of Cimpri\v c regarding $*$-orderability of quantum
groups.
Keywords:$*$--orderings, valuations, rings with involution Categories:14P10, 16S30, 16W10 |
4. CMB 2003 (vol 46 pp. 14)
Generalized Commutativity in Group Algebras We study group algebras $FG$ which can be graded by a finite abelian
group $\Gamma$ such that $FG$ is $\beta$-commutative for a
skew-symmetric bicharacter $\beta$ on $\Gamma$ with values in $F^*$.
Categories:16S34, 16R50, 16U80, 16W10, 16W55 |
5. 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 |