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.
holds. is said to be irreducible if it is not reducible.
of irreducible subsets of .
We define the Krull dimension of a ring as the dimension of .