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A Sheaf Representation of Principally Quasi-Baer ∗-Rings

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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.

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Acknowledgments

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

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Authors and Affiliations

  1. Department of Mathematics, Abasaheb Garware College, Pune, 411004, India

    Anil Khairnar

  2. Center For Advanced Studies in Mathematics, Department of Mathematics, Savitribai Phule Pune University, Pune, 411007, India

    B. N. Waphare

Authors
  1. Anil Khairnar
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  2. B. N. Waphare
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Correspondence to Anil Khairnar.

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Presented by Yuri Drozd.

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Khairnar, A., Waphare, B.N. A Sheaf Representation of Principally Quasi-Baer ∗-Rings. Algebr Represent Theor 22, 79–97 (2019). https://doi.org/10.1007/s10468-017-9758-0

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  • DOI: https://doi.org/10.1007/s10468-017-9758-0

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