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.