0

Definition of valuation ring:
Let $R$ be an integral domain with $frac(R)=K$. Then $R$ is said to be a valuation ring if

(1) $R \neq K$

(2) $\forall x \in K, x \in R$ or $x^{-1} \in R$.

Now my problem is:

Let $A, B$ be two valuation rings with $K=frac(A)=frac(B)$. Now suppose $A⊂B$, can we conclude that $A=B$?

I can prove that this is not true if $A$ is a discrete valuation ring:

For suppose the maximal ideal in $A$ is generated by $t$, then the only non units in $A$ are of the form $ut^{n}$, with $u$ a unit in $A$, $n$ an integer $\geq 1$.

If $A \subsetneq B$, then $B$ must contains the inverse of some element of the above form, i. e. $u^{-1}t^{-n}=(ut^n)^{-1} \in B$, but then $ \frac 1 t = (u t^{n-1})(u^{-1}t^{-n}) \in B$. Hence $B\supset A[\frac 1 t ]= K$.

However, I can't think of anything useful in the general case that $A, B$ may not be discrete.

Any help will be appreciated.

user26857
  • 52,094
chan kifung
  • 1,302

1 Answers1

3

If $A$ is a valuation ring with quotient field $K$, then any ring $B$ between $A$ and $K$ is a valuation ring.

D_S
  • 33,891
  • Oh, so now the question becomes: If A is a valuation ring, can we find another ring C such that $A\subsetneq C \subsetneq K$, I believe it is not true, but I cannot think of an example – chan kifung Mar 21 '16 at 03:24
  • Do you know of any non-discrete valuation rings you could play around with? – D_S Mar 21 '16 at 03:45
  • Hint: there exist valuation rings with finite Krull dimension greater than one. Let $A$ be such a ring. Then there exists a prime ideal $\mathfrak p$ of $A$ which is not zero and which is not the maximal ideal of $A$. What can you do with $\mathfrak p$? – D_S Mar 21 '16 at 03:50
  • So we would localize $A$ at $\mathfrak p$, I have found on the Internet that the ring $k[x,x/y,x/y^2,\cdots ]_{(y)}$ is an example of valuation ring with krull dimension $>1$, thank you so much – chan kifung Mar 21 '16 at 04:23
  • You're welcome. Actually your question got me thinking about very similar question which I just asked (http://math.stackexchange.com/questions/1706740/localization-of-a-valuation-ring-at-a-prime-is-abstractly-isomorphic-to-the-orig). I'm too tired to do it now, but maybe you can solve it before I can :) – D_S Mar 21 '16 at 04:28