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A Sheaf Representation of Principally Quasi-Baer ∗-Rings

  • Anil Khairnar
  • B. N. Waphare
Article

Abstract

The concept of a central strict ideal in a principally quasi-Baer (p.q.-Baer) ∗-ring is introduced. It is proved that the set of all prime central strict ideals in a p.q.-Baer ∗-ring is an anti-chain with respect to set inclusion. We obtain a separation theorem, which ensures an existence of prime central strict ideals in a p.q.-Baer *-ring. It is proved that the set of all prime central strict ideals (not necessarily prime ideals) of a p.q.-Baer ∗-ring carries the hull-kernel topology. We investigate the Hausdorffness and the compactness of this topology. As an application of spectral theory, it is proved that p.q.-Baer ∗-rings have a sheaf representation with injective sections. The class of p.q.-Baer ∗-rings which have a sheaf representation with stalks to be p.q.-Baer ∗-rings is determined.

Keywords

p.q.-Baer ∗-rings Central strict ideals Hull-kernel topology Sheaf representation 

Mathematics Subject Classification (2010)

Primary 16S60 Secondary 16W10 16D70 

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Notes

Acknowledgments

The authors are thankful to the anonymous referees for helpful comments and suggestions.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAbasaheb Garware CollegePuneIndia
  2. 2.Center For Advanced Studies in Mathematics, Department of MathematicsSavitribai Phule Pune UniversityPuneIndia

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