# An ascending chain of prime ideals.

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.

• $(0)\subset(X_1)\subset(X_1,X_2)\subset(X_1,X_2,X_3)\subset(X_1,X_2,X_3,X_4)$ has five terms. Apr 17, 2019 at 13:10

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.

• In $K[X_1,\dots,X_n,\dots]$ consider $P_i=(X_i,\dots)$. These form a descending chain. And consider $Q_i=(X_,\dots,X_i)$. These form an ascending chain. Apr 17, 2019 at 13:09
• And btw, Nagata's example doesn't (and can't) contain an infinite chain of ascending prime ideals. And any noetherian ring doesn't. Apr 17, 2019 at 13:15
• @user26857 damn. you are right, I will make a clear explaining soon. Feeling sorry for making poor quality answers. Apr 17, 2019 at 15:56