Nonzero ideals in a discrete valuation ring

Question

We say that $K$ is a discrete valuation field if there exists a surjective map $v:K\to\mathbb{Z}\cup\{\infty\}$ such that

1. $v(xy)=v(x)+v(y)$
2. $v(x+y)=\inf\{v(x),v(y)\}$
3. $v(x)=\infty$ if and only if $x=0$

We then define the discrete valuation ring to be $A=\{x\in K\mid v(x)\geq 0\}$.

I am trying to show that if $A$ is a discrete valuation ring, and $m=(\pi)$ with $v(\pi)=1$, then all ideals in $A$ are of the form $(\pi^n)$ for some $n$. Now one way to do this is to show that $A$ is noetherian and to invoke Krull's intersection theorem, but neither have been discussed in class, so I'm wondering: is there a way to show this directly?

Answer 1

Hint:

This results from the fact that if $x\in A$ is such that $v(x)=n>0$, then we can write $$x=u\pi^n,\quad\text{where }u \text{ is a unit in }A.$$