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:
[1] | Aghajani, M.; Tarizadeh, A., Characterizations of Gelfand rings specially clean rings and their dual rings, Results Math., 75, 3, 125 (2020) · Zbl 1448.14001 |
[2] | Bezhanishvili, G., Stone duality and Gleason covers through de Vries duality, Topol. Appl., 157, 6, 1064-1080 (2010) · Zbl 1190.54015 |
[3] | Borceux, F.; Janelidze, G., Galois Theories (2001), Cambridge University Press · Zbl 0978.12004 |
[4] | de Jong, A. J., The Stacks Project (2021), see |
[5] | Gleason, A. M., Projective topological spaces, Ill. J. Math., 2, 4A, 482-489 (1958) · Zbl 0083.17401 |
[6] | Givant, S.; Halmos, P. R., Introduction to Boolean Algebras (2008), Springer |
[7] | Halmos, P. R., Lectures on Boolean Algebras (1963), D. Van Nostrand Company, Inc.: D. Van Nostrand Company, Inc. New Jersey · Zbl 0114.01603 |
[8] | Hindman, N.; Strauss, D., Algebra in the Stone-Čech Compactification, Theory and Applications (2012), De Gruyter Textbook · Zbl 1241.22001 |
[9] | Hochster, M., Prime ideal structure in commutative rings, Trans. Am. Math. Soc., 142, 43-60 (1969) · Zbl 0184.29401 |
[10] | Johnstone, P. T., Stone Spaces (1982), Cambridge University Press · Zbl 0499.54001 |
[11] | Sabogal P., S. M., An extension of the Stone duality: the expanded version, Bol. Mat., 13, 1, 1-19 (2006) · Zbl 1203.06014 |
[12] | Semandeni, Z., Banach Spaces of Continuous Functions, vol. I (1971), Polish Scientific Publishers: Polish Scientific Publishers Warsaw · Zbl 0225.46030 |
[13] | Stone, M. H., The theory of representations of Boolean algebras, Trans. Am. Math. Soc., 40, 37-111 (1936) · JFM 62.0033.04 |
[14] | Strauss, D. P., Extremally disconnected spaces, Proc. Am. Math. Soc., 18, 2, 305-309 (1967) · Zbl 0147.41302 |
[15] | Tarizadeh, A., Flat topology and its dual aspects, Commun. Algebra, 47, 1, 195-205 (2019) · Zbl 1410.13001 |
[16] | Tarizadeh, A., On the category of profinite spaces as a reflective subcategory, Appl. Gen. Topol., 14, 2, 147-157 (2013) · Zbl 1323.54018 |
[17] | Walker, R. C., The Stone-Čech Compactification (1974), Springer · Zbl 0292.54001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.