Stone type representations and dualities by power set ring. (English) Zbl 1462.06004
Summary: In this paper, it is shown that the Boolean ring of a commutative ring is isomorphic to the ring of clopens of its prime spectrum. In particular, Stone’s Representation Theorem is generalized. The prime spectrum of the Boolean ring of a given ring \(R\) is identified with the Pierce spectrum of \(R\). The discreteness of prime spectra is characterized. It is also proved that the space of connected components of a compact space \(X\) is isomorphic to the prime spectrum of the ring of clopens of \(X\). As another major result, it is shown that a morphism of rings between complete Boolean rings preserves suprema if and only if the induced map between the corresponding prime spectra is an open map.
MSC:
| 06E15 | Stone spaces (Boolean spaces) and related structures |
| 14A05 | Relevant commutative algebra |
| 54B30 | Categorical methods in general topology |
| 12F10 | Separable extensions, Galois theory |
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
| 06E05 | Structure theory of Boolean algebras |
| 06E20 | Ring-theoretic properties of Boolean algebras |
Keywords:
power set ring; Boolean ring; clopen; complete Boolean ring; Galois connection; stone type dualityReferences:
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