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# Local ring

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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.