Let be a field, a discrete valuation on , and the valuation ring of . For each integer , define .
(a) Prove that for any , is a principal ideal, and that
(b) Prove that if is any nonzero ideal of , then for some .
three things that you need to recall first:
1) since is surjective, for some which is clearly in because
2) if and only if is a unit: this is a quick result of this fact that
3) for some if and only if for sime unit : this is an immediate result of 2).
now we want to show that is a principal ideal. first from 1) we have: thus hence suppose now that and then
hence by 3): this proves that and part (a) of your problem is solved.
for part (b), let obviously choose with then and so for some unit therefore