In $K[X,Y]$, is the power of any prime also primary?

Question

I've recently been reading about primary decomposition, and was browsing the questions here. From this, I know that it is not true that every primary ideal is the power of a prime ideal.

I'm curious about a slight variation inspired by the comments.

Consider $K[X,Y]$, for $K$ a field. For any prime $\mathfrak{p}\subset K[X,Y]$, and any $n\geq 1$, is the ideal $\mathfrak{p}^n$ primary, so that if $ab\in\mathfrak{p}^n$, then either $a\in\mathfrak{p}^n$ or $b^j\in\mathfrak{p}^n$ for some $j$?

I suspect it is indeed true, and it is obviously true when $n=1$. An inductive approach doesn't feel right, so is there another way to reach the conclusion?

score 10 · Accepted Answer · edited Oct 05 '16 at 21:02

10

The answer is yes. Recall the following important fact:

Theorem. (Reid, Undergraduate Commutative Algebra, Proposition page 22) The prime ideals of $k[x,y]$ are as follows:

$0$;
$(f)$, for irreducible $f \in k[x,y]$;
maximal ideals $\mathfrak{m}$.

Now you can easily conclude with the following two facts: (a) a power of a maximal ideal is primary (Atiyah, Macdonald 4.2); (b) in a UFD the power of a principal prime ideal is primary.

I don't know what happens in general in $k[x_1, \dots, x_n]$, when $n \geq 3$. It can be proved that if $\mathfrak{p}$ is a monomial prime in $k[x_1, \dots, x_n]$ then $\mathfrak{p}^m$ is $\mathfrak{p}$-primary for all $m$.

edited Oct 05 '16 at 21:02

user26857

52,094

answered Feb 24 '12 at 19:36

Andrea

7,606

I doubt this holds for $n\geq 3$. – Alex Becker Feb 24 '12 at 20:27
Thanks Andrea. Do you have a proof or reference to a proof of the second fact (b) stated above? – Kally Feb 25 '12 at 01:16
It is quite easy. Try to prove it on your own. – Andrea Feb 25 '12 at 13:46
@Alex: you are right to doubt! See a non-primary square of a prime in Atiyah-Macdonald, page 51, Example 3). – Georges Elencwajg Sep 03 '15 at 06:42
@GeorgesElencwajg That doubt refers to the polynomial ring over a field in (at least) three variables. There is such an example in Northcott, Ideal Theory, page 29, Example 3. – user26857 Oct 05 '16 at 21:06
dear Andrea, +1, but perhaps it is also worth nothing that point (b) holds in any integral domain, not just UFDs :) ... (ie a power of a principal prime ideal in an integral domain is always primary) – Atticus Stonestrom Jun 17 '21 at 18:24

In $K[X,Y]$, is the power of any prime also primary?

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