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Complete local rings as domains

Published online by Cambridge University Press:  12 March 2014

V. Stoltenberg-Hansen  and
J. V. Tucker
V. Stoltenberg-Hansen
Affiliation:
Department of Mathematics, Uppsala University, S-752 38 Uppsala, Sweden
J. V. Tucker
Affiliation:
Centre for Theoretical Computer Science, University of Leeds, Leeds 1S2 9TJ, England
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Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.

Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra  and an embedding i: A →  are constructed such that  is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.

The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure  is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field of recursive real numbers.

In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.

A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt . Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.

Type
Survey/expository papers
Information
The Journal of Symbolic Logic , Volume 53 , Issue 2 , June 1988 , pp. 603 - 624
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[1] Atiyah, M. F. and Macdonald, I. G., Introduction to commutative algebra, Addison-Wesley, Reading, Massachusetts, 1969.Google Scholar
[2] de Bakker, J. W. and Zucker, J. I., Processes and the denotational semantics of concurrency, Information and Control, vol. 54 (1982), pp. 70–120.CrossRefGoogle Scholar
[3] Baumslag, G., Cannonito, F. B., and Miller, C. F. III, Computable algebra and group embeddings, Journal of Algebra, vol. 69 (1981), pp. 186–212.CrossRefGoogle Scholar
[4] Baur, W., Rekursive Algebren mit Kettenbedingungen, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 20 (1974), pp. 37–46.CrossRefGoogle Scholar
[5] Bergstra, J. A. and Klop, J. W., Process algebra for synchronous communication, Information and Control, vol. 60 (1984), pp. 109–137.CrossRefGoogle Scholar
[6] Bourbaki, N., Elements of mathematics. General topology. Part I, Addison-Wesley, Reading, Massachusetts, 1966.Google Scholar
[7] Ershov, Y. L., Model C of partial continuous functionals, Logic Colloquium '76 (Gandy, R. and Hyland, M., editors), North-Holland, Amsterdam, 1977, pp. 455–467.Google Scholar
[8] Fröhlich, A. and Shepherdson, J. C., Effective procedures in field theory, Philosophical Transactions of the Royal Society of London, Series A, vol. 248 (1956), pp. 407–432.Google Scholar
[9] Hermann, G., Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Mathematische Annalen, vol. 95 (1926), pp. 736–788.CrossRefGoogle Scholar
[10] Hingston, P., Effective decomposition in Noetherian rings, Aspects of effective algebra (Crossley, J. N., editor), Upside Down A Book Company, Yarra Glen, 1981, pp. 122–127.Google Scholar
[11] Lachlan, A. H. and Madison, E. W., Computable fields and arithmetically definable ordered fields, Proceedings of the American Mathematical Society, vol. 24 (1970), pp. 803–807.CrossRefGoogle Scholar
[12] Lazard, D., Algorithmes fondamentaux en algèbre commutative, Journées arithmétiques (Paris, 1975), Astérisque, vol. 38/39, Société Mathématique de France, Paris, 1976, pp. 131–138.Google Scholar
[13] Mal'cev, A. I., Constructive algebras. I, The metamathematics of algebraic systems: collected papers 1936–1967, North-Holland, Amsterdam, 1971, pp. 148–212.Google Scholar
[14] Matsumura, H., Commutative algebra, 2nd ed., Benjamin/Cummings, Reading, Massachusetts, 1980.Google Scholar
[15] Rabin, M. O., Computable algebra, general theory and the theory of computable fields, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
[16] Richman, F., Constructive aspects of Noetherian rings, Proceedings of the American Mathematical Society, vol. 44 (1974), pp. 436–441.CrossRefGoogle Scholar
[17] Scott, D. S., Lectures on a mathematical theory of computation, Oxford University Computing Laboratory Programming Research Group Technical Monograph PRG-19, Oxford, 1981; reprinted in Theoretical foundations of programming methodology (M. Broy and D. Schmidt, editors), Reidel, Dordrecht, 1982, pp. 145–292.Google Scholar
[18] Seidenberg, A., Constructions in algebra, Transactions of the American Mathematical Society, vol. 197 (1974), pp. 273–313.CrossRefGoogle Scholar
[19] Seidenberg, A., What is Noetherian? Rendiconti del Seminario Matematico e Fisico di Milano, vol. 44 (1974), pp. 55–61.CrossRefGoogle Scholar
[20] Sigstam, I., Effective aspects of Noetherian rings and modules, Project Report No. 1, Department of Mathematics, Uppsala University, Uppsala, 1983.Google Scholar
[21] Stoltenberg-Hansen, V. and Tucker, J. V., Computable algebra. An introduction to computable rings and fields (in preparation).Google Scholar
[22] Stoltenberg-Hansen, V. and Tucker, J. V., Algebraic and fixed-point equations over inverse limits of algebras, Report 11.87, Centre for Theoretical Computer Science, University of Leeds, Leeds, 1987.Google Scholar
[23] Suter, G. H., Recursive elements and constructive extensions of computable local integral domains, this Journal, vol. 38 (1973), pp. 272–290.Google Scholar