Local ring: Difference between revisions
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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. | ||
The [[localisation (ring theory)|localisation]] of a [[commutativity|commutative]] [[integral domain]] at a non-zero [[prime ideal]] is a local ring. | |||
==Properties== | |||
In a local ring the unit group is the [[complement]] of the maximal ideal. | |||
==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=100,206-207 }}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:01, 12 September 2024
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.
The localisation of a commutative integral domain at a non-zero prime ideal is a local ring.
Properties
In a local ring the unit group is the complement of the maximal ideal.
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, 100,206-207. ISBN 0-201-55540-9.