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Noetherian orders

Published online by Cambridge University Press:  02 December 2010

HERVÉ PERDRY  and
PETER SCHUSTER
HERVÉ PERDRY
Affiliation:
Université Paris-Sud UMR-S 669 and Inserm U 669, 16 avenue Paul-Vaillant-Couturier, Bâtiment Inserm 15/16, Villejuif F-94807
PETER SCHUSTER
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
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Abstract

Noether classes of posets arise in a natural way from the constructively meaningful variants of the notion of a Noetherian ring. Using an axiomatic characterisation of a Noether class, we prove that if a poset belongs to a Noether class, then so does the poset of the finite descending chains. When applied to the poset of finitely generated ideals of a ring, this helps towards a unified constructive proof of the Hilbert basis theorem for all Noether classes.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Issue 1 , February 2011 , pp. 111 - 124
Copyright
Copyright © Cambridge University Press 2010

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References

Berger, J., Ishihara, H. and Schuster, P. (2009) The weak König lemma, Brouwer's fan theorem, De Morgan's law, and dependent choice (preprint).Google Scholar
Coquand, T. and Lombardi, H. (2006) A logical approach to abstract algebra. Mathematical Structures in Computer Science 16 885–900.Google Scholar
Edwards, H. M. (2005) Essays in Constructive Mathematics, Springer-Verlag.Google Scholar
Jacobsson, C. and Löfwall, C. (1991) Standard bases for general coefficient rings and a new constructive proof of Hilbert's basis theorem. J. Symb. Comput. 12 (3)337–372.Google Scholar
Lombardi, H. and Quitté, C. (to appear) Algèbre commutative. Méthodes constructives. Modules projectifs de type fini, Calvage and Mounet. (English version forthcoming at Springer-Verlag; preliminary version available at the home page of H. Lombardi.)Google Scholar
Mines, R., Richman, F. and Ruitenburg, W. (1988) A Course in Constructive Algebra, Universitext, Springer-Verlag.Google Scholar
Perdry, H. (2004) Strongly noetherian rings and constructive ideal theory. J. Symb. Comput. 37 (4)511–535.CrossRefGoogle Scholar
Perdry, H. (2008) Lazy bases: a minimalist constructive theory of Noetherian rings. Math. Log. Quart. 54 (1)70–82.Google Scholar
Richman, F. (1974) Constructive aspects of Noetherian rings. Proc. Amer. Math. Soc. 44 436–441.CrossRefGoogle Scholar
Richman, F. (2003) The ascending tree condition: constructive algebra without countable choice. Comm. Algebra 31 1993–2002.CrossRefGoogle Scholar
Schuster, P. and Zappe, J. (2006) Do Noetherian rings have Noetherian basis functions? In: Beckmann, A., Berger, U., Löwe, B. and Tucker, J. V. (eds.) Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006, Swansea, UK, July 2006. Springer-Verlag Lecture Notes in Computer Science 3988 481–489.Google Scholar
Seidenberg, A. (1974) What is Noetherian? Rend. Sem. Mat. Fis. Milano 44 55–61.Google Scholar
Simpson, S. G. (1988) Ordinal numbers and the Hilbert basis theorem. J. Symb. Logic 53 (3)961–974.Google Scholar
Simpson, S. G. (1999) Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag.Google Scholar
Tennenbaum, J. (1973) A Constructive Version of Hilbert's Basis Theorem, Ph.D. thesis, University of California San Diego.Google Scholar
Zariski, O. and Samuel, P. (1958) Commutative Algebra, Volume I, Van Nostrand.Google Scholar