Citizendium - building a quality free general knowledge encyclopedia. Click here to join and contribute—free Many thanks December donors; special to Darren Duncan. January donations open; need minimum total \$100. Let's exceed that. Donate here. By donating you gift yourself and CZ.

# Local ring

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

A ring $A$ is said to be a local ring if it has a unique maximal ideal $m$. 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 $\bigcap_n m^n = \{0\}$ and A is complete with respect to the uniformity defined by the cosets of the powers of m.