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 .

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- December 8th 2008, 08:02 PMxianghu21valuation ring questions
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 . - December 9th 2008, 09:26 AMNonCommAlg
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