If $R$ is a DVR then we know that $R$ has only two prime ideals. Does this still hold true for valuation ring $R$?
I was trying to prove this by showing: Let $R$ be a valuation ring. I wanted to prove given $x,y \in R$ if $0 < v(x) < v(y)$ then $v(x^m)> v(y)$ for some $m \in \mathbb{Z}$.
But as I have been pointed out by the comments this does not hold in general.
Are there examples of valuation ring $R$ with more than two prime ideals? Any comments/examples would be appreciated. Thank you.
ps I have changed my question from asking how to show the above inequality to this since the question I was asking is not true in general as pointed out in the comments.