By using the Gauss's lemma, we see that every PID is normal.
Normalizations are useful to reduce singularities.
and denote the class of in by respectively. is not normal. Indeed, satisfies a monic equation
Thus the normalization of contains the element . Now, let us note that equation
holds so that holds. Since is normal, we see that . Note that is not locally free whereas is free.
In other words, a normalization of a ring can never be flat (unless the trivial case where itself is normal).
For any subset of we define
Then we may topologize in such a way that the closed sets are sets of the form for some . Namely,
We refer to the topology as the Zariski topology.
It is continuous with respect to the Zariski topology.
is a closed map with respect to the Zariski topology.