# Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 03.1   For any commutative ring , we define its spectrum as

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 Zariski topology.

LEMMA 03.2   For any ring , the following facts holds.
1. For any subset of , we have

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

PROPOSITION 03.3   For any ring homomorphism , we have a map

It is continuous with respect to the Zariski topology.

PROPOSITION 03.4   For any ring , the following statements hold.
1. For any ideal of , let us denote by the canonical projection. Then gives a homeomorphism between and .
2. For any element of , let us denote by be the canonical map. Then gives a homeomorphism between and .

PROPOSITION 03.5   Let be a ring. Let be a ring homomoprhism. We regard as an module via If is a finite -module, then

is a closed map with respect to the Zariski topology.

DEFINITION 03.6   Let be a topological space. A closed set of is said to be reducible if there exist closed sets and such that

holds. is said to be irreducible if it is not reducible.

DEFINITION 03.7   Let be an ideal of a ring . Then we define its radical to be

such that

PROPOSITION 03.8   Let be a ring. Then;
1. For any ideal of , we have .
2. For two ideals , of , holds if and only if .
3. For an ideal of , is irreducible if and only if is a prime ideal.

DEFINITION 03.9   We define a dimension of a topological space as a supremum of the length of sequences

of irreducible subsets of .

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