Maximal ideal of a local ring

Let $R$ be a local ring . Its defining feature is that it has a unique maximal ideal , usually denoted $\mathfrak m$.

Statement

If $(R,\mathfrak m)$ is a local ring, then:

1. $\mathfrak m$ is the unique maximal ideal of $R$.
2. An element $u\in R$ is a unit iff $u\notin \mathfrak m$. Equivalently, $R^\times = R\setminus \mathfrak m.$
3. The Jacobson radical of $R$ equals $\mathfrak m$: $J(R)=\mathfrak m.$
4. Via the residue field $k=R/\mathfrak m$, an element $u\in R$ is a unit iff its image in $k$ is nonzero.

Examples

1. $\mathbb Z_{(p)}$.
   In the local ring $\mathbb Z_{(p)}$, the maximal ideal is $p\mathbb Z_{(p)}$. The units are fractions $a/b$ with $p\nmid a$.

2. $k[[x]]$.
   In k[[x]] , the maximal ideal is $(x)$. A power series is a unit iff its constant term is nonzero.

3. $k[x,y]_{(x,y)}$.
   The localization of $k[x,y]$ at the maximal ideal $(x,y)$ is local with maximal ideal generated by $x$ and $y$.