Abstract
Let κ be a cardinal number. If κ ≥ 2, then there exists a (commutative unital) ring A such that the set of A-algebra isomorphism classes of minimal ring extensions of A has cardinality κ. The preceding statement fails for κ = 1 and, if A must be nonzero, it also fails for κ = 0. If \( \kappa \leq \aleph _{0} \), then there exists a ring whose set of maximal (unital) subrings has cardinality κ. If an infinite cardinal number κ is of the form κ = 2λ for some (infinite) cardinal number λ, then there exists a field whose set of maximal subrings has cardinality κ.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Azarang, A.: On maximal subrings. Far East J. Math. Sci. 32, 107–118 (2009)
Azarang, A., Karamzadeh, O.A.S.: On the existence of maximal subrings in commutative Artinian rings. J. Algebra Appl. 9, 771–778 (2010)
Azarang, A., Karamzadeh, O.A.S.: Which fields have no maximal subrings? Rend. Sem. Mat. Univ. Padova 126, 213–228 (2011)
Azarang, A., Karamzadeh, O.A.S.: On maximal subrings of commutative rings. Algebra Colloq. 19, 1125–1138 (2012)
Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading, MA (1969)
Dobbs, D.E.: Cech cohomology and a dimension theory for commutative rings. Ph.D. thesis, Cornell University, Ithaca, New York (1969)
Dobbs, D.E.: Every commutative ring has a minimal ring extension. Commun. Algebra 34, 3875–3881 (2006)
Dobbs, D.E.: On the commutative rings with at most two proper subrings. Int. J. Math. Math. Sci. 2016, 1–13 (2016). Article ID 6912360. doi:10.1155/2016/6912360
Dobbs, D.E., Mullins, B.: On the lengths of maximal chains of intermediate fields in a field extension. Commun. Algebra 29, 4487–4507 (2001)
Dobbs, D.E., Shapiro, J.: A classification of the minimal ring extensions of certain commutative rings. J. Algebra 308, 800–821 (2007)
Dobbs, D.E., Mullins, B., Picavet, G., Picavet-L’Hermitte, M.: On the FIP property for extensions of commutative rings. Commun. Algebra 33, 3091–3119 (2005)
Ferrand, D., Olivier, J.-P.: Homomorphismes minimaux d’anneaux. J. Algebra 16, 461–471 (1970)
Gilmer, R.: Multiplicative Ideal Theory. Dekker, New York (1972)
Halmos, P.R.: Naive Set Theory. Van Nostrand, Princeton (1960)
Huckaba, J.A.: Commutative Rings with Zero Divisors. Dekker, New York (1988)
Hungerford, T.W.: Algebra. Springer, New York (1974)
Jacobson, N.: Lectures in Abstract Algebra, Volume III - Theory of Fields and Galois Theory. Van Nostrand, Princeton (1964)
Kaplansky, I.: Commutative Rings, rev. ed. University of Chicago Press, Chicago (1974)
Lewis, W.J.: The spectrum of a ring as a partially ordered set. J. Algebra 25, 419–434 (1973)
Picavet, G., Picavet-L’Hermitte, M.: About minimal morphisms. In: Multiplicative Ideal Theory in Commutative Algebra, pp. 369–386. Springer, New York (2006)
Picavet, G., Picavet-L’Hermitte, M.: Modules with finitely many submodules. Int. Electron. J. Algebra 19, 119–131 (2016)
Stepräns, J.: The number of submodules. Proc. Lond. Math. Soc. 49, 183–192 (1984)
Werner, N.J.: Covering numbers of finite rings. Am. Math. Mon. 122, 552–566 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Dobbs, D.E. (2017). Commutative Rings with a Prescribed Number of Isomorphism Classes of Minimal Ring Extensions. In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules. Springer, Cham. https://doi.org/10.1007/978-3-319-65874-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-65874-2_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65872-8
Online ISBN: 978-3-319-65874-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)