{{Short description|Mathematical concept in dimension theory of local rings}} {{More citations needed|date=May 2022}} In [[mathematics]], a '''system of parameters''' for a [[local ring|local]] [[Noetherian ring]] of [[Krull dimension]] ''d'' with [[maximal ideal]] ''m'' is a set of elements ''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub> that satisfies any of the following equivalent conditions: # ''m'' is a [[Minimal prime ideal|minimal prime]] over (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>). # The [[radical of an ideal|radical]] of (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) is ''m''. # Some power of ''m'' is contained in (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>). # (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) is [[primary ideal|''m''-primary]]. # R/(''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) is an [[Artinian ring]]. Every local Noetherian ring admits a system of parameters.<ref name=Hochster>{{cite web |url=https://web.archive.org/web/20241231053848/https://dept.math.lsa.umich.edu/~hochster/711F07/L09.05.pdf | title=Math 711: Lecture of September 5, 2007 | publisher=University of Michigan| date=September 5, 2007 | access-date=March 9, 2026}}</ref>
It is not possible for fewer than ''d'' elements to generate an ideal whose radical is ''m'' because then the dimension of ''R'' would be less than ''d''.
If ''M'' is a ''k''-dimensional module over a local ring, then ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub> is a '''system of parameters''' for ''M'' if the [[Length of a module|length]] of {{nowrap|''M'' / (''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) ''M''}} is finite.
==General references== *{{Citation |last1=Atiyah |first1=Michael Francis |title=[[Introduction to Commutative Algebra]] |year=1969 |publisher=Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. |mr=0242802 |last2=Macdonald |first2=I. G. |author1-link=Michael Atiyah |author2-link=Ian G. Macdonald}} *{{Citation |last1=Eisenbud |first1=David |title=Commutative Algebra |year=1995 |publisher=Springer-Verlag, New York |mr=1322960|author1-link=David Eisenbud}}
==References== {{reflist}}
[[Category:Commutative algebra]] [[Category:Ideals (ring theory)]]
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