Local ring: Difference between revisions
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imported>David E. Volk No edit summary |
imported>Richard Pinch (supplied References Lang, section anchor Complete local ring) |
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A ring <math>A</math> is said to be a '''local ring''' if it has a unique maximal ideal <math>m</math>. It is said to be ''semi-local'' if it has finitely many maximal ideals. | A ring <math>A</math> is said to be a '''local ring''' if it has a unique maximal ideal <math>m</math>. It is said to be ''semi-local'' if it has finitely many maximal ideals. | ||
==Complete local ring== | |||
A local ring ''A'' is '''complete''' if the intersection <math>\bigcap_n m^n = \{0\}</math> and ''A'' is complete with respect to the [[uniformity]] defined by the cosets of the powers of ''m''. | |||
==References== | |||
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=100 }} | |||
Revision as of 11:08, 21 December 2008
A ring is said to be a local ring if it has a unique maximal ideal . It is said to be semi-local if it has finitely many maximal ideals.
Complete local ring
A local ring A is complete if the intersection and A is complete with respect to the uniformity defined by the cosets of the powers of m.
References
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 100. ISBN 0-201-55540-9.