Proposition 9

Let $A$ be a local integral domain distinct from its field of fractions. The following conditions are equivalent: (a) $A$ is the ring of a discrete valuation. (b) $A$ is a principal ideal domain. (c) The ideal $m(A)$ is principal and $\bigcap_{n=1}^{\infty} m(A)^n = (0)$.

(d) $A$ is a Noetherian ring and $m(A)$ is principal. (e) $A$ is a Noetherian valuation ring.

Proposition 8 shows that (a) implies (b), (d) and (e). If $A$ is a principal ideal domain, then $m(A) = Au$ and every non-zero ideal of $A$ is of the form $Au^n$ since $A$ is local (Algebra, Chapter VII, § 1, no. 3, Theorem 2); therefore $\bigcap_{n=1}^\infty m(A)^n = 0$; this shows that (b) implies (c). On the other hand (d) implies (c) (Chapter III, § 3, no. 2, Corollary to Proposition 5); by Proposition 2 of § 1, no. 4, (c) implies (a). Thus conditions (a), (b), (c), (d) are equivalent and imply (e). Finally suppose (e) holds and let us show that (b) holds; it will be sufficient to prove the following lemma: