A Sheaf Representation of Principally Quasi-Baer ∗-Rings
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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.
Keywordsp.q.-Baer ∗-rings Central strict ideals Hull-kernel topology Sheaf representation
Mathematics Subject Classification (2010)Primary 16S60 Secondary 16W10 16D70
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The authors are thankful to the anonymous referees for helpful comments and suggestions.
- 1.Berberian, S.K.: Baer ∗-Rings, Grundlehren Math. Wiss. Band 195, vol. 296. Springer, Berlin (1972)Google Scholar
- 6.Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Principally quasi-Baer rings hulls, Advances in Ring Theory Trends in Mathematics, 47-61, Birkhäuser Basel (2010)Google Scholar
- 8.Dauns, J., Hofmann, K.H.: Representation of rings by sections. Mem. Amer. Math. Soc. 83 (1968)Google Scholar
- 16.Keimel, K.: Lecture notes in mathematics 248, pp 1–98. Springer-Verlag, Berlin (1971)Google Scholar
- 17.Kim, J.Y.: On reflexive principally quasi-Baer rings. Korean J. Math. 17(3), 233–236 (2009)Google Scholar
- 18.Khairnar, A., Waphare, B.N.: Unitification of weakly p.q.-Baer ∗-rings, (to appear in Southeast Asian Bull. Math.)Google Scholar