## Journal of Commutative Algebra

### A constructive theory of minimal zero-dimensional extensions

Fred Richman

#### Abstract

Chiorescu characterized the minimal zero-dimensional extensions of certain one-dimensional rings in terms of families of ideals indexed by prime ideals. In this paper we give a constructive development of these extensions, which, to achieve maximum generality, must necessarily avoid dependence on prime ideals. This forces us to develop a purely arithmetic theory. Along the way we get a characterization, in terms of the lattice of radicals of finitely generated ideals, of when a ring with primary zero-ideal has dimension at most one.

#### Article information

Source
J. Commut. Algebra, Volume 5, Number 4 (2013), 545-566.

Dates
First available in Project Euclid: 31 January 2014

https://projecteuclid.org/euclid.jca/1391192656

Digital Object Identifier
doi:10.1216/JCA-2013-5-4-545

Mathematical Reviews number (MathSciNet)
MR3161746

Zentralblatt MATH identifier
1291.13016

Subjects
Primary: 13B02: Extension theory

#### Citation

Richman, Fred. A constructive theory of minimal zero-dimensional extensions. J. Commut. Algebra 5 (2013), no. 4, 545--566. doi:10.1216/JCA-2013-5-4-545. https://projecteuclid.org/euclid.jca/1391192656

#### References

• Miroslav Arapović, The minimal $0$-dimensional overrings of commutative rings, Glas. Mat. 18 (1983), 47-52.
• James W. Brewer and Fred Richman, Subrings of zero-dimensional rings, in Multiplicative ideal theory in commutative algebra, Springer, New York, 2006, 73-88.
• Marcela Chiorescu, Minimal zero-dimensional extensions, J. Alg. 322 (2009), 259-269.
• Thierry Coquand, Lionel Ducos, Henri Lombardi and Claude Quitté, Constructive Krull dimension I: Integral extensions, J. Alg. Appl. 8 (2009), 129-138.
• Thierry Coquand, Henri Lombardi and Marie-Françoise Roy, An elementary characterization of Krull dimension, From sets and types to topology and analysis, Oxford Logic Guides 48 (2005), 239-244.
• Lionel Ducos, Henri Lombardi, Claude Quitté and Maimouna Salou, Théorie algorithmique des anneaux arithmétiques, des anneaux de Prüfer et des anneaux de Dedekind, J. Alg. 281 (2004), 604-650.
• Ray Mines, Fred Richman and Wim Ruitenburg, A course in constructive algebra, Springer, New York, 1988.
• Jack Ohm and Robert Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968), 631-640. \noindentstyle