Abstract view
Published:20100726
Printed: Dec 2010
Gary F. Birkenmeier,
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, U.S. A.
Jae Keol Park,
Department of Mathematics, Busan National University, Busan, South Korea
S. Tariq Rizvi,
Department of Mathematics, Ohio State University, Lima, OH, U.S.A.
Abstract
We investigate the behavior of the quasiBaer and the
right FIextending 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 quasiBaer 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 quasiBaer 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 torsionfree Abelian group $G$
over a commutative semiprime quasicontinuous 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
(FI)extending, Morita equivalent, ring of quotients, essential overring, (quasi)Baer ring, ring hull, u.p.monoid, $C^*$algebra

MSC Classifications: 
16N60, 16D90, 16S99, 16S50, 46L05 show english descriptions
Prime and semiprime rings [See also 16D60, 16U10] Module categories [See also 16Gxx, 16S90]; module theory in a categorytheoretic context; Morita equivalence and duality None of the above, but in this section Endomorphism rings; matrix rings [See also 15XX] General theory of $C^*$algebras
16N60  Prime and semiprime rings [See also 16D60, 16U10] 16D90  Module categories [See also 16Gxx, 16S90]; module theory in a categorytheoretic context; Morita equivalence and duality 16S99  None of the above, but in this section 16S50  Endomorphism rings; matrix rings [See also 15XX] 46L05  General theory of $C^*$algebras
