Douady and Douady's "Algèbre et théories galoisiennes" proposes an exercise where we must show that different definitions of a discrete valuation ring are equivalent (3.1.10.b):
Let $A$ be a ring and $w : A \to \mathbb{N} \cup \{\infty\}$ be a function verifying the following properties:
- $w(xy) = w(x) + w(y)$;
- $w(x + y) \geq \min(w(x), w(y))$;
- $w(x) < \infty$ for $x \neq 0$;
- $w(x) > 0$ for a non-invertible $x$.
Show that $A$ is a principal ring with a unique maximal ideal.
This seems a bit different from other definitions that I've seen (such as the one on Wikipedia), which state that:
A discrete valuation ring is an integral domain $A$ such that there exists a function $w : F \to \mathbb{Z} \cup \{\infty\}$, where $F$ is the field of fractions of $A$, such that
- $w(xy) = w(x) + w(y)$;
- $w(x + y) \geq \min(w(x), w(y))$;
- $w(x) = \infty \iff x = 0$;
- $A = \{ x \in F \mid w(x) \geq 0 \}$.
Are these two sets of conditions really equivalent? In particular, the second set implies that, if $x$ and $y \neq 0$ are elements of $A$ such that $w(y) \leq w(x)$, then $x/y \in A$, whereas I don't see how this could hold under the first set.
