A constructive theory of minimal zero-dimensional extensions. (English) Zbl 1291.13016

Richman, Fred

J. Commut. Algebra 5, No. 4, 545-566 (2013).

The author proves a constructive version of M. Chiorescu’s theorem [J. Algebra 322, No. 1, 259–269 (2009; Zbl 1173.13007)] that gives a complete set of invariants for minimal zero-dimensional extensions of a commutative ring \(R\) which satisfies the three conditions:
1. \(\dim R \leq 1\),
2. The zero ideal of \(R\) is primary,
3. \(R\) has a Noetherian spectrum.
Her characterization was given in terms of families of ideals indexed by prime ideals. The present author avoids dependence on prime ideals in order to achieve maximum generality in developing constructively these extensions. To that end, a purely arithmetic theory is developed. Among numerous results that this paper offers, one can find a characterization of the condition that a ring with primary zero-ideal has dimension at most one, in terms of the lattice of radicals of finitely generated ideals. Titles of the paper’s sections show further the content and the development of the ideas as follows: 2. Krull dimension zero; 3. The invariant; 4. The lattice \(L(R)\); 5. Extending admissible families; 6. Algebraic extensions of factorial fields; 7. Arapović’s theorem; 8. The main theorem for suitable rings; 9. Noetherian spectrum; 10. A one-dimensional Bézout domain.

Reviewer: Radoslav M. Dimitrić (New York)

MSC:

13B02	Extension theory of commutative rings
03F65	Other constructive mathematics
03D45	Theory of numerations, effectively presented structures

Keywords:

zero-dimensional extensions; constructive theory; Noetherian spectrum; one-dimensional rings; arithmetic theory; Krull dimension; suitable rings; regular ideal; factorial field; Bézout domain

Citations:

Zbl 1173.13007

References:

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