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1. 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 |
2. CMB 2010 (vol 53 pp. 223)
Density of Polynomial Maps Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.
Keywords:density, polynomial, endomorphism ring, PI Categories:16D60, 16S50 |
3. CMB 2009 (vol 52 pp. 145)
$2$-Clean Rings A ring $R$ is said to be $n$-clean if every
element can be written as a sum of an idempotent and $n$ units.
The class of these rings contains clean rings and $n$-good rings
in which each element is a sum of $n$ units. In this paper, we
show that for any ring $R$, the endomorphism ring of a free
$R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$
of all $\omega\times \omega$ row and column-finite matrices over
any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is
considered where $R$ is a local ring.
Keywords:$2$-clean rings, $2$-good rings, free modules, row and column-finite matrix rings, group rings Categories:16D70, 16D40, 16S50 |
4. CMB 2006 (vol 49 pp. 265)
Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in
C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$-clean if every element is
the sum of a unit and a root of $g(x)$. If $V$ is a vector space of
countable dimension over a division ring $D,$ they showed that
$\end {}_{D}V$ is
$g(x)$-clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=x-x^{2}$
this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan.
In this paper we remove the countable condition, and in fact prove that
$\Mend {}_{R}M$ is $g(x)$-clean for any semisimple module $M$ over an arbitrary
ring $R$ provided that $g(x)\in (x-a)(x-b)C[x]$ where $a,b\in C$ and both $b$
and $b-a$ are units in $R$.
Keywords:Clean rings, linear transformations, endomorphism rings Categories:16S50, 16E50 |
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 1997 (vol 40 pp. 198)
The ${\cal J}_0$-radical of a matrix nearring can be intermediate An example is constructed to show that the ${\cal J}_0$-radical of a matrix
nearring can be an intermediate ideal. This solves a conjecture put forward
in [1].
Categories:16Y30, 16S50, 16D25 |