Theorem 1

Let K be a field and V a subring of K. The following conditions are equivalent: (a) V is a maximal element of the set of local subrings of K, this set being ordered by the relation "B dominates A" between A and B. (b) There exists an algebraically closed field L and a homomorphism h from V to L which is maximal in the set of homomorphisms from subrings of K to L, ordered by the relation "g is an extension off" between f and g. (c) If $x \in K - V$, then $x^{-1} \in V$. (d) The field of fractions of V is K and the set of principal ideals of V is totally ordered by the relation of inclusion. (e) The field of fractions of V is K and the set of ideals of V is totally ordered by the relation of inclusion.

We shall show the theorem by proving the following implications

Suppose that (a) holds. Then $V$ is a local ring. Let $L$ be an algebraic closure of the residue field $\kappa(V)$ and $h$ the canonical homomorphism from $V$ to $L$. Let $V'$ be a subring of $K$ containing $V$ and $h'$ a homomorphism from $V'$ to $L$ extending $h$. If $p'$ is the kernel of $h'$, then $p' \cap V = m(V)$; hence (no. 1, Example 2) $V'_{p'}$ dominates $V$, which implies $V'_{p'} = V$ and $V' = V$. Thus (b) holds.

Suppose (b) holds. Let $L$ be an algebraically closed field and $h$ a homomorphism from $V$ to $L$; suppose that $h$ is maximal in the set of homomorphisms from subrings of $K$ to $L$; let $p$ be the kernel of $h$. The elements of $h(V - p)$ being invertible in $L$, $h$ can be extended to a homomorphism from $V_p$ to $L$ (Chapter II, § 2, no. 1, Proposition 1); hence $V = V_p$, which shows that $V$ is a local ring and that $p$ is its maximal ideal. Let $x$ be a non-zero element of $K$; we must show that one at least of the elements $x, x^{-1}$ belongs to $V$, that is, by virtue of the maximal character of $h$, that $h$ can be extended to $V[x]$ or $V[x^{-1}]$. If $x$ is integral over $V$, this follows from Chapter V, § 2, no. 1, Corollary 4 to Theorem 1. If $x$ is not integral over $V$, we shall use the following lemma: