Let $A$ be a DVR in its field of fractions $F$, and F'\subset F a subfield. Then is it true that A\cap F' is a DVR in F'? I can see that it is a valuation ring of F', but how do I I show, for example, that A\cap F' is Noetherian?
Sub-discrete valuation ring?
2 Answers
Restrict the valuation corresponding to A to F'. Then the restriction is also a valuation with discreate value group( a subgroup of ring of integers) if it is non trivial. But if the value group is trivial, then it is field and therefore not a DVR.
This is not necessarily true. For instance, you could have $A = k[[t]]$ for a field $k$ and F' \subset k((t)) equal to $k$. Then A \cap F' = F' and a field is often not considered a DVR.
But this is the only thing that can go wrong. In general, if a domain has a nontrivial valuation taking values in the nonnegative integers, such that the elements of valuation zero are units, then it is a DVR (and noetherianness follows --- for instance, one checks that the only ideals occur as ${x: x \ \mathrm{has \ valuation \ at \ least \ } n}.$)
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0@Pete: Dear Pete, OK. I am simply following the conventions on Wikipedia. I've made a note in the answer, though. – 2011-02-27