Definition 3

Let $A$ be a ring and $p$ a prime ideal of $A$. The ring $A$, is called the local ring of $A$ at $p$, or the local ring of $p$, when there is no ambiguity.

Remark (2). If $A$ is a local ring and $m$ its maximal ideal, the elements of $A - m$ are invertible (Proposition 1) and hence $A_{m'}$ is canonically identified with $A$ ($§ 2$, no. 1, Remark 5).

Examples (5) Let $p$ be a prime number. The local ring $\mathbf{Z}_{(p)}$ is the set of rational numbers $a/b$, where $a, b$ are rational integers with $b$ prime top; the residue field of $\mathbf{Z}_{(p)}$ is isomorphic to the prime field $\mathbf{F}_p = \mathbf{Z}/(p)$.

(6) Let $V$ be an affine algebraic variety, $A$ the ring of regular functions on $V$, $W$ an irreducible subvariety of $V$ and $p$ the (necessarily prime) ideal of $A$ consisting of the functions which are zero at every point of $W$. The ring $A$, is called the local ring of $W$ on $V$. \*