Expand all Collapse all | Results 1 - 10 of 10 |
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 2012 (vol 56 pp. 584)
On Automorphisms and Commutativity in Semiprime Rings Let $R$ be a semiprime ring with center
$Z(R)$. For $x,y\in R$, we denote by $[x,y]=xy-yx$ the commutator of
$x$ and $y$. If $\sigma$ is a non-identity automorphism of $R$ such
that
$$
\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0
$$
for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed
positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$
such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when
$R$ is a prime ring, $R$ is commutative.
Keywords:automorphism, generalized polynomial identity (GPI) Categories:16N60, 16W20, 16R50 |
3. CMB 2010 (vol 53 pp. 587)
Hulls of Ring Extensions We investigate the behavior of the quasi-Baer and the
right FI-extending right ring hulls under various ring extensions
including group ring extensions, full and triangular matrix ring
extensions, and infinite matrix ring extensions. As a consequence,
we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are
Morita equivalent, then so are the quasi-Baer right ring hulls
$\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of
$R$ and $S$, respectively. As an application, we prove that if
unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings,
then the bounded central closure of $A$ and that of $B$ are
strongly Morita equivalent as $C^*$-algebras. Our results show
that the quasi-Baer property is always preserved by infinite
matrix rings, unlike the Baer property. Moreover, we give an
affirmative answer to an open question of Goel and Jain for the
commutative group ring $A[G]$ of a torsion-free Abelian group $G$
over a commutative semiprime quasi-continuous ring $A$. Examples
that illustrate and delimit the results of this paper are provided.
Keywords:(FI-)extending, Morita equivalent, ring of quotients, essential overring, (quasi-)Baer ring, ring hull, u.p.-monoid, $C^*$-algebra Categories:16N60, 16D90, 16S99, 16S50, 46L05 |
4. CMB 2005 (vol 48 pp. 355)
On Maps Preserving Products Maps preserving certain algebraic properties of elements
are often studied in Functional Analysis and Linear Algebra. The
goal of this paper is to discuss the relationships among these
problems from the ring-theoretic point of view.
Categories:16W20, 16N50, 16N60 |
5. CMB 2000 (vol 43 pp. 413)
Non-Isomorphic Maximal Orders with Isomorphic Matrix Rings We construct a countably infinite family of pairwise non-isomorphic
maximal ${\mathbb Q}[X]$-orders such that the full $2$ by $2$
matrix rings over these orders are all isomorphic.
Categories:16S50, 16H05, 16N60 |
6. 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 |
7. CMB 1999 (vol 42 pp. 174)
Rings With Comparability The class of rings studied in this paper properly contains the
class of right distributive rings which have at least one
completely prime ideal in the Jacobson radical. Amongst other
results we study prime and semiprime ideals, right noetherian rings
with comparability and prove a structure theorem for rings with
comparability. Several examples are also given.
Categories:16U99, 16P40, 16D14, 16N60 |
8. CMB 1998 (vol 41 pp. 452)
Dependent automorphisms in prime rings For each $n\geq 4$ we construct a class of examples of a minimal
$C$-dependent set of $n$ automorphisms of a prime ring $R$, where $C$
is the extended centroid of $R$. For $n=4$ and $n=5$ it is shown that
the preceding examples are completely general, whereas for $n=6$ an
example is given which fails to enjoy any of the nice properties of
the above example.
Categories:16N60, 16W20 |
9. CMB 1998 (vol 41 pp. 79)
An answer to a question of Kegel on sums of rings We construct a ring $R$ which is a sum of two subrings
$A$ and $B$ such that the Levitzki radical of $R$ does not
contain any of the hyperannihilators of $A$ and $B$. This
answers an open question asked by Kegel in 1964.
Categories:16N40, 16N60 |
10. CMB 1998 (vol 41 pp. 81)
The cardinality of the center of a $\PI$ ring The main result shows that if $R$ is a semiprime ring satisfying
a polynomial identity, and if $Z(R)$ is the center of $R$, then
$\card R \leq 2^{\card Z(R)}$. Examples show that this bound can
be achieved, and that the inequality fails to hold for rings which
are not semiprime.
Categories:16R20, 16N60, 16R99, 16U50 |