# Local ring

From Citizendium, the Citizens' Compendium

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 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 <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

- Serge Lang (1993).
*Algebra*, 3rd ed. Addison-Wesley, 100,206-207. ISBN 0-201-55540-9.