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Local ring
From Citizendium, the Citizens' Compendium
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.