Residue field

Let $(R,\mathfrak m)$ be a local ring (so $\mathfrak m$ is its unique maximal ideal ).

Definition

The residue field of $(R,\mathfrak m)$ is the quotient ring

Because $\mathfrak m$ is maximal, $k$ is a field (see also quotient ring ).

For a localization at a prime ideal $R_{\mathfrak p}$, the residue field is often denoted

Useful observation

An element $u\in R$ is a unit iff its image $\bar u\in k$ is nonzero. Equivalently, $u$ is a unit iff $u\notin\mathfrak m$.

Examples

1. Integers localized at $p$.
   For $R=\mathbb Z_{(p)}$, the maximal ideal is $p\mathbb Z_{(p)}$, so

   $$R/\mathfrak m \cong \mathbb F_p.$$

2. Formal power series.
   For $R=k[[x]]$ with $\mathfrak m=(x)$,

   $$k[[x]]/(x)\cong k.$$

3. A local ring at a point.
   Let $R=k[x,y]_{(x-a,y-b)}$. Then the residue field is $k$ (the images of $x,y$ become $a,b$).