Suppose $(R,v)$ is a valuation ring with a value group which is a subgroup of the real numbers under addition. Now there are two natural ways to make $R$ into a topological ring:
- Fix some real number $r > 1$ and define a metric on $R$ via $d(x,y) = r^{-v(x-y)}$
- As $R$ is local, there is a unique maximal ideal $\mathfrak{m}$, so we can consider the $\mathfrak{m}$-adic topology.
Now the obvious question is how do these topologies relate to each other, in particular: do they coincide? Note that if $R$ is a DVR, this is easy, because we have $\mathfrak{m}^n = \{x \in R | v(x) \geq n\}$, but this method does not generalize.
In general the answer is "No": lets assume that the value group of $v$ has no minimal positive element. Then for $r\in m\setminus 0$ there exists $s\in m\setminus 0$ such that $v(s)<v(r)$. Hence $r=s\cdot\frac{r}{s}$, where $\frac{r}{s}\in m$. This shows $m^2=m$ and therefore $m^k=m$ for all $k\in\mathbb{N}$. Consequently the $m$-adic topology is not Hausdorff.