Example 5

Let $p$ be a prime number. We denote by $\mathbf{Z}_{(p)}$ the subring of the field $\mathbf{Q}$ of rational numbers consisting of the fractions $a/b$ with $a\in \mathbf{Z},b\in \mathbf{Z}$, and $b$ not divisible by $p*$(cf. Comm. Alg., II, §2, No. 1, p. 60) . Then $\mathbf{Z}$ is

$*(_{p)}$

a commutative local ring, with maximal ideal $p\mathbf{Z}_{(p)}$. The ring $\mathbf{Z}_p$ of $p$-adic integers (V, §12, No. 3, p. 96) is a commutative local ring, with maximal ideal $p\mathbf{Z}_p$ (VIII, p. 40, Exercise 9).