Proposition 2

Let $A$ be a ring and $p$ a prime ideal of $A$. The ring $A$, is local; its maximal ideal is the ideal $pA_p = p_v$, generated by the canonical image of $p$ in $A$; its residue field is canonically isomorphic to the field of fractions of $A/p$.

Let $S = A - p$ and $j : A \to A_v$, be the canonical homomorphism; the hypothesis that $p$ is prime implies that $p$ is saturated with respect to $S$, hence $j^{-1}(pA_p) = p$ ($§ 2$, no. 4, Proposition 10) and, as the ideals of $A$ not meeting $S$ are those contained in $p$, the first two assertions are special cases of $§ 2$, no. 5, Proposition 11 (ii). Moreover, iff is the canonical homomorphism $A \to A/p$, $f(S)$ is the set of elements $\neq 0$ of the integral domain $A/p$ and hence the last assertion is a special case of $§ 2$, no. 5, Proposition 11 (i).