Let
be a ring homomorphism. Let
and
. If
, then
is a prime ideal of
, i.e., a point of
. Hence
induces a map
. Show that:
If
is surjective, then
is a homeomorphism of Y onto the closed subset
of .
[
denotes the set of all prime ideals of
which contain
].
Here is one way to do it:
By the "Ideal Correspondence Theorem," we know that and induces a bijective map from to . Then I need to show
is continuous and
is continuous to show it is a homeomorphism.
I don't understand the sentence in red. How do we know
induces a bijective map from to ?