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BOUNDS FOR INDEXES OF NILPOTENCY IN COMMUTATIVE RING THEORY: A PROOF MINING APPROACH

Part of: Proof theory and constructive mathematics General commutative ring theory Ring extensions and related topics

Published online by Cambridge University Press:  05 January 2021

FERNANDO FERREIRA
FERNANDO FERREIRA*
Affiliation:
DEPARTAMENTO DE MATEMÁTICA FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA CAMPO GRANDE, ED. C6, 1749-016LISBOA, PORTUGAL E-mail: [email protected]
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Abstract

It is well-known that an element of a commutative ring with identity is nilpotent if, and only if , it lies in every prime ideal of the ring. A modification of this fact is amenable to a very simple proof mining analysis. We formulate a quantitative version of this modification and obtain an explicit bound. We present an application. This proof mining analysis is the leitmotif for some comments and observations on the methodology of computational extraction. In particular, we emphasize that the formulation of quantitative versions of ordinary mathematical theorems is of independent interest from proof mining metatheorems.

Keywords

Proof miningquantitative versionsnilpotencycommutative ringsbounded functional interpretation

MSC classification

Primary: 03F10: Functionals in proof theory 13A99: None of the above, but in this section
Secondary: 13B25: Polynomials over commutative rings 03F35: Second- and higher-order arithmetic and fragments
Type
Communication
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 3-4 , December 2020 , pp. 257 - 267
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Copyright
© The Association for Symbolic Logic 2021

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