2019/05/25 by Tarizadeh, Abolfazl, Taheri, Zahra
#06E05 #06E15 #06E20 #13A15 #13C05 #13M05 #14A05 #54H25 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.10612
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.