Stone type representations and dualities by power set ring

[摘要] 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. (C) 2021 Elsevier B.V. All rights reserved.

[发布日期] 2021-11-01

[关键词] Power set ring;Boolean ring;Clopen;Complete Boolean ring;Galois connection;Stone type duality