Local ring

A commutative ring $R$ is a local ring if it has a unique maximal ideal. One often records this ideal and writes $(R,\mathfrak m)$, where $\mathfrak m$ is the unique maximal ideal.

The unique maximal ideal of a local ring is closely tied to units; see maximal ideal of a local ring . The quotient $R/\mathfrak m$ is the residue field of $R$.

Equivalent characterizations

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

1. $R$ is local (i.e. it has a unique maximal ideal).
2. The set of nonunits in $R$ is an ideal; this ideal is then the unique maximal ideal.
3. Whenever $a+b=1$ in $R$, at least one of $a$ or $b$ is a unit.

Local rings arise systematically from localization: if $\mathfrak p$ is a prime ideal of $R$, then localizing at the prime produces the local ring $R_{\mathfrak p}$.

Many foundational results in commutative algebra are naturally stated for local rings; for instance, Nakayama's lemma is formulated for finitely generated modules over a local ring.

Examples

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

2. Localizing $\mathbb Z$ at a prime. For a prime number $p$, the ring $\mathbb Z_{(p)}$ from localization at (p) is local, with maximal ideal $p\mathbb Z_{(p)}$.

3. Localizing a polynomial ring at a maximal ideal. If $k$ is a field, then $k[x]_{(x)}$ is local with maximal ideal generated by $x$. More generally, $k[x,y]_{(x,y)}$ is local with maximal ideal $(x,y)$.