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A sheaf representation of principally quasi-Baer \(\ast\)-rings. (English) Zbl 1408.16022

Summary: The concept of a central strict ideal in a principally quasi-Baer (p.q.-Baer) \(\ast\)-ring is introduced. It is proved that the set of all prime central strict ideals in a p.q.-Baer \(\ast\)-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 \(\ast\)-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 \(\ast\)-rings have a sheaf representation with injective sections. The class of p.q.-Baer \(\ast\)-rings which have a sheaf representation with stalks to be p.q.-Baer \(\ast\)-rings is determined.

MSC:

16S60 Associative rings of functions, subdirect products, sheaves of rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16U99 Conditions on elements
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