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Hulls of Ring Extensions

  Published:2010-07-26
 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.
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Abstract

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 (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 category-theoretic context; Morita equivalence and duality
None of the above, but in this section
Endomorphism rings; matrix rings [See also 15-XX]
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 category-theoretic context; Morita equivalence and duality
16S99 - None of the above, but in this section
16S50 - Endomorphism rings; matrix rings [See also 15-XX]
46L05 - General theory of $C^*$-algebras
 

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