Zariski topology

Question

Is there any relation between the Krull intersection theorem and Zariski topology? If it exists? How? I searched for this relationship but I can't find any articles.

-Thanks if you can help me.

Answer 1

As the wikipedia article on completion of a ring says:

In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull topology (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal $I=\mathfrak {m}$ is especially important, for example the distinguished maximal ideal of a valuation ring. The basis of open neighbourhoods of $0$ in $R$ is given by the powers $I_n$, which are nested and form a descending filtration on $R$.

[The canonical map from the ring into the completion] is injective if and only if this intersection reduces to the zero element of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is either an integral domain or a local ring.

You can read on to see the Krull topology for modules too. So this is the right topology link you are looking for.

The Zariski topology OTOH, I believe, is tied more to localizations rather than completions. Localizing and completion are two very important but very different operations in commutative algebra.