**Yoshifumi Tsuchimoto**

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
.

holds. is said to be

such that

- For any ideal of , we have .
- For two ideals , of , holds if and only if .
- For an ideal of , is irreducible if and only if is a prime ideal.

of irreducible subsets of .

We define the Krull dimension of a ring as the dimension of .