Regular ring: Difference between revisions
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In mathematics, a '''regular ring''' is a [[Noetherian ring]] such that the [[localisation]] at every [[prime ideal]] is a [[Regular Local Ring|regular local ring]]. | {{subpages}} | ||
In mathematics, a '''regular ring''' is a [[Noetherian ring]] such that the [[localisation]] at every [[prime ideal]] is a [[Regular Local Ring|regular local ring]]: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its [[Krull dimension]]. | |||
[[Jean-Pierre Serre]] defines a regular ring as a commutative noetherian ring of finite [[global homological dimension]] and shows that this is equivalent to the definition above. For regular rings, Krull dimension agrees with global homological dimension. | |||
Examples of regular rings include fields (of dimension zero) and [[Dedekind domain]]s. If ''A'' is regular then so is ''A''[''X''], with dimension one greater than that of ''A''. | |||
==See also== | |||
* [[von Neumann regular ring]], a different concept with a similar name. | |||
==References== | |||
* [[Jean-Pierre Serre]], ''Local algebra'', [[Springer-Verlag]], 2000, ISBN 3-540-66641-9. Chap.IV.D. | |||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:CZ Live]] | [[Category:CZ Live]] | ||
Revision as of 15:09, 30 October 2008
In mathematics, a regular ring is a Noetherian ring such that the localisation at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.
Jean-Pierre Serre defines a regular ring as a commutative noetherian ring of finite global homological dimension and shows that this is equivalent to the definition above. For regular rings, Krull dimension agrees with global homological dimension.
Examples of regular rings include fields (of dimension zero) and Dedekind domains. If A is regular then so is A[X], with dimension one greater than that of A.
See also
- von Neumann regular ring, a different concept with a similar name.
References
- Jean-Pierre Serre, Local algebra, Springer-Verlag, 2000, ISBN 3-540-66641-9. Chap.IV.D.