What are the properties of valuation rings and their ideals?

[mathsss2]

Let $$K$$ be a field, $$\nu : K^* \rightarrow \texbb{Z}$$ a discrete valuation on $$K$$, and $$R=\{x \in K^* : \nu(x) \geq 0 \} \cup \{0\}$$ the valuation ring of $$\nu$$. For each integer $$k \geq 0$$, define $$A_k=\{r \in R : \nu(r) \geq k \} \cup \{0\}$$.

(a) Prove that for any $$k$$, $$A_k$$ is a principal ideal, and that $$A_0 \supseteq A_1 \supseteq A_2 \supseteq\ldots$$
(b) Prove that if $$I$$ is any nonzero ideal of $$R$$, then $$I=A_k$$ for some $$k \geq 0$$.

 

[mathwonk]

the only tools you have are facts about integers (well ordering) and the definition of a valuation. try looking at an element of minimal valuation.