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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:

  1. $w(xy) = w(x) + w(y)$;
  2. $w(x + y) \geq \min(w(x), w(y))$;
  3. $w(x) < \infty$ for $x \neq 0$;
  4. $w(x) > 0$ 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

  1. $w(xy) = w(x) + w(y)$;
  2. $w(x + y) \geq \min(w(x), w(y))$;
  3. $w(x) = \infty \iff x = 0$;
  4. $A = \{ x \in F \mid w(x) \geq 0 \}$.

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.

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Let $A=k[[t^2,t^3]]\subset B=k[[t]]$, where $k$ is a field. Then $B$ is a dvr and the associated valuation induces a map on $A$ which satisfies your properties 1-4, but $A$ is not a pid.

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