Residue field

Let $(R,\mathfrak m)$ be a local ring. The residue field of $R$ is the quotient ring

Because $\mathfrak m$ is maximal (see maximal ideal of a local ring), the quotient $R/\mathfrak m$ is a field.

The canonical surjection

is sometimes called the residue map.

More generally, for a prime ideal $\mathfrak p\subset R$ one often sets

where $R_{\mathfrak p}$ is the localization at p; this recovers the same construction after passing to that local ring.

Examples

1. $\mathbb Z_{(p)}$. For the local ring $\mathbb Z_{(p)}$, the maximal ideal is $p\mathbb Z_{(p)}$, and
   $$\mathbb Z_{(p)}/p\mathbb Z_{(p)}\cong \mathbb F_p.$$

1. $k[x]_{(x)}$. In $R=k[x]_{(x)}$, the maximal ideal is $(x)R$, so
   $$k[x]_{(x)}/(x)\ \cong\ k$$
   via evaluation at $x=0$.

1. $k[x,y]_{(x,y)}$. In the local ring $k[x,y]_{(x,y)}$, the maximal ideal is $(x,y)$, and the residue field is again $k$.