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Asymptotic Differential Algebra and Model Theory of Transseries$
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Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Print publication date: 2017

Print ISBN-13: 9780691175423

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691175423.001.0001

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Some Commutative Algebra

Some Commutative Algebra

Chapter:
(p.29) Chapter One Some Commutative Algebra
Source:
Asymptotic Differential Algebra and Model Theory of Transseries
Author(s):

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691175423.003.0002

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring R and an R-module M, with U and V as additive subgroups of R and M. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.

Keywords:   commutative algebra, Johnson's Theorem, algebraic differential equation, regular local ring, Kähler differentials, commutative ring, Zariski topology, noetherian ring, integrally closed domain, Krull's Principal Ideal Theorem

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