Maximal ideal of a local ring

For $R_{\mathfrak p}$, the localization at a prime, the unique maximal ideal is $\mathfrak pR_{\mathfrak p}$, and the associated residue field is $R_{\mathfrak p}/\mathfrak pR_{\mathfrak p}$.

This units-versus-maximal-ideal dichotomy is used in Nakayama's lemma.

1. $\mathbb Z_{(p)}$. In $R=\mathbb Z_{(p)}$, a fraction $\frac{a}{b}$, where $p\nmid b$, is a unit if and only if $p\nmid a$. Thus the maximal ideal is $p\mathbb Z_{(p)}$.

1. $k[x]_{(x)}$. In $R=k[x]_{(x)}$, the units are exactly the fractions $\frac{f}{g}$ with $f(0)\neq 0$. Hence the maximal ideal is generated by $x$.

1. A field. In a field $k$, the only nonunit is $0$, so $\mathfrak m=(0)$.