View source for Approximation property (ring theory)

In [[Abstract algebra|algebra]], a commutative [[Noetherian ring]] ''A'' is said to have the '''approximation property''' with respect to an [[Ideal (ring theory)|ideal]] ''I'' if each finite system of polynomial equations with coefficients in ''A'' has a solution in ''A'' if and only if it has a solution in the ''I''-adic completion of ''A''.<ref name=rotthaus/><ref name=stacks/> The notion of the approximation property is due to [[Biography:Michael Artin|Michael Artin]].

== See also ==
*[[Artin approximation theorem]]
*[[Popescu's theorem]]

== Notes ==
{{Reflist|refs=

<ref name=rotthaus>{{Cite journal |jstor=44238106 |doi=10.1216/rmjm/1181071964|title=Excellent Rings, Henselian Rings, and the Approximation Property|journal=Rocky Mountain Journal of Mathematics|volume=27|issue=1|pages=317–334|year=1997|last1=Rotthaus|first1=Christel|doi-access=free}}</ref>

<ref name=stacks>{{Cite web|url=https://stacks.math.columbia.edu/tag/07BW|title=Tag 07BW: Smoothing Ring Maps|last=|first=|date=|website=The Stacks Project|publisher=[[Organization:Columbia University|Columbia University]], Department of Mathematics|access-date=2018-02-19}}</ref>

}}

==References==
*{{cite journal|last=Popescu |first=Dorin |title=General Néron desingularization and approximation |journal=Nagoya Mathematical Journal |year=1986 |volume=104 |pages=85&ndash;115 |doi=10.1017/S0027763000022698 |url=https://projecteuclid.org/download/pdf_1/euclid.nmj/1118780554|doi-access=free }}
*{{cite journal |last=Rotthaus |first=Christel |title=On the approximation property of excellent rings |journal=[[Inventiones Mathematicae]] |year=1987 |volume=88 |pages=39&ndash;63 |doi=10.1007/BF01405090 }}
*{{cite journal |doi=10.1007/BF02684596 |title=Algebraic approximation of structures over complete local rings|journal=Publications Mathématiques de l'IHÉS|volume=36|pages=23–58|year=1969|last1=Artin|first1=M|issn=0073-8301|url=http://www.numdam.org/item/PMIHES_1969__36__23_0/}}
*{{cite journal |doi=10.1007/BF01389777 |title=On the solutions of analytic equations|journal=Inventiones Mathematicae|volume=5|issue=4|pages=277–291|year=1968|last1=Artin|first1=M|issn=0020-9910}}

[[Category:Ring theory]]



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