## Valuation Rings, Discrete Valuation Ring

### Discrete Valuation Ring

Let the ring R be a valuation ring, with valuation group G.
Remember that G is linearly ordered.
We can assign the elements of G the
order topology.
If this topology is discrete
(I'll explain this below), then R is a discrete valuation ring,
also known as a dvr.
Remember that every point in a discrete topology is an open set.
In other words, every x has an immediate predecessor and successor.
These two elements bound x and make x an open set.

If you think about it, it's hard to build a valuation group that is *not* discrete.
Here's why.
Suppose G is not discrete at x.
This means x is a cluster point of G.
Consider any other point y and add y-x to all the points in G.
This transformation preserves order, so y is also a cluster point.
Since y was arbitrary, every point in G is a cluster point.

There are ordered groups that look like this, but they usually aren't valuation groups.
Consider the group generated by 1 and sqrt(2).
This is isomorphic to **Z*****Z**, with the ordering of the reals.
The group is countable, so it doesn't cover all the reals.
If a/b is a rational number that is within ε of the square root of 2,
then a-b×sqrt(2) is within ε of 0.
This makes 0 a cluster point,
and every point is a cluster point.
In fact, G is dense in the real line.
Clearly G does not have the discrete topology,
but G isn't a valuation group either.

If R is a valuation ring and a pid, it produces a valuation group equal to **Z**.
Every element of G has an immediate predecessor and successor,
G is discrete,
and R is a dvr.