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Hulls of Ring Extensions

Published online by Cambridge University Press:  20 November 2018

Gary F. Birkenmeier ,
Jae Keol Park  and
S. Tariq Rizvi
Gary F. Birkenmeier
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, U.S. A. e-mail: [email protected]
Jae Keol Park
Affiliation:
Department of Mathematics, Busan National University, Busan, South Korea e-mail: [email protected]
S. Tariq Rizvi
Affiliation:
Department of Mathematics, Ohio State University, Lima, OH, U.S.A. e-mail: [email protected]
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Abstract

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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 ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(R)$ and ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(S)$ 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

16N6016D9016S9916S5046L05(FI-) extending, Morita equivalentring of quotientsessential overring(quasi-) Baer ringring hullu.p.-monoidC*-algebra
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 4 , 01 December 2010 , pp. 587 - 601
Copyright
Copyright © Canadian Mathematical Society 2010

References

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