Local ring

A commutative ring $R$ is called a local ring if it has a unique maximal ideal . When the unique maximal ideal is $\mathfrak m$, one often writes the pair as $(R,\mathfrak m)$.

Equivalent characterizations

For a commutative ring $R$, the following are equivalent:

• $R$ has a unique maximal ideal $\mathfrak m$.
• The set of non-units in $R$ is an ideal (necessarily the unique maximal ideal).
  (See maximal ideal of a local ring .)

A local ring has an associated residue field $R/\mathfrak m$.

Examples

1. Fields.
   Any field is local: its unique maximal ideal is $(0)$.

2. Localization at a prime ideal.
   For any prime $\mathfrak p\subset R$, the localization $R_{\mathfrak p}$ is a local ring, with maximal ideal $\mathfrak pR_{\mathfrak p}$.

3. Formal power series.
   The ring k[[x]] is local, with maximal ideal $(x)$ and residue field $k$.