Proposition 8

Let $K$ be a (not necessarily commutative) field, $v$ a discrete valuation on $K$, $A$ the ring of $v$ and $u$ a uniformizer for $v$. The non-zero ideals of $A$ are two-sided and of the form $Au^n$ ($n \geq 0$).

It may be assumed that $v$ is normed, so that $v(u) = 1$. For all $x \in K^*$, there is an integer $n \in \mathbf{Z}$ such that $(v(x) = n = v(u^n)$ and hence we may write

where $z, z'$ are two invertible elements of the ring $A$; whence the proposition.