**Yoshifumi Tsuchimoto**

- is integral over .
- is a finite -module.
- There exists a subring of which contains as a subset such that is a finite module over .

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).

**

is a prime ideal of $A$

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,

closed

We refer to the topology as the

- For any subset
of
, we have
- For any subset
of
, let us denote by
the ideal of
generated by
. then we have

- For any ideal of , let us denote by the canonical projection. Then gives a homeomorphism between and .
- For any element
of
, let us denote by
be the canonical map. Then
gives a homeomorphism between
and
.