An ascending chain of prime ideals.

Question

I am trying to get used to $\operatorname{Spec}$ of a ring. I know an example, when one prime ideal is contained in another for $\mathbb{C}[x,y]$. $(f) \subset (x-a,y-b)$, where $f(a,b) = 0$.

Is there, say, an example of ascending chain of prime ideals exactly $5$ terms long?

Could you give an example of an infinite ascending chain of prime ideals?

The same question for an infinite descending chain of prime ideals?

I think that I want to see an explicit example of usage of going up/down theorem, something useful is discussed here.

Answer 1

The answer can be found googling "Krull Dimension" and "Nagata example of an Krull infinitely dimensional Noetherian ring." The non-intuitive though conceptual answer is found.

Furthermore, the question about the descending chain of ideals is still to answer.

UPD: the ideal $I$ is called prime if $ab \in I$ implies $a \in I$ or $b \in I$

Lemma ideal $I = (x_1, x_2 ... x_n) $ , where $x_i$ are distinct variables of $\mathbb{C}[x_1, x_2 ... ]$ is prime.

Justification: if $f*g$ is in $I$, then it should be divisible by a polynomial of first $n$ variables. If neither $f$ nor $g$ of them are, then $fg$ is not divisible also.

That's because complex polynomial ring is a UFD

This gives us the way to construct an ascending chain of prime ideals of any length.

Lemma 2 $(x_2 ... x_n ...)$ is a prime ideal for the same reason. This gives us a descending chain of prime ideals.

The ring of polynomials with infinitely many variables is not noetherian by definition, though.