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Duality on compact prime ringed spaces. (English) Zbl 0821.16045

The author studies conditions under which a ring \(R\) is the ring of global sections of some compact ringed space \((X,{\mathfrak F})\) in the sense of C. J. Mulvey [J. Algebra 52, 411-436 (1978; Zbl 0418.18009)]. He introduces the notion of a small weakly Baer ring which generalizes the notion of a Baer ring.
The paper contains the following results: 1) A ring \(R\) is isomorphic to the ring of global sections of a compact ringed space \((X, {\mathfrak F})\) for which each stalk of \(\mathfrak F\) is a prime ring if and only if \(R\) is a small weakly Baer ring. 2) The category of small Baer rings is dual to the category of compact prime ringed spaces. 3) The ring of global sections of a ringed space \((X, {\mathfrak F})\) is a Baer ring if and only if \(X\) is a Stone space and each stalk of \(\mathfrak F\) is a domain. 4) The category of Baer rings is dual to the category of compact domain ringed spaces.

MSC:

16W80 Topological and ordered rings and modules
16S60 Associative rings of functions, subdirect products, sheaves of rings
16D90 Module categories in associative algebras
16D25 Ideals in associative algebras
06F25 Ordered rings, algebras, modules

Citations:

Zbl 0418.18009
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