6
$\begingroup$

Is there a finite dimensional local ring with infinitely many minimal prime ideals?

Equivalent formulation:

Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of minimal prime sub-ideals of $\mathfrak p$ is infinite?


Here "ring" means "commutative ring with one", "dimension" means "Krull dimension", and "local ring" means "ring with exactly one maximal ideal" (warning: some authors call "quasi-local ring" a ring with exactly one maximal ideal, and "local ring" a noetherian ring with exactly one maximal ideal; it is well known that a noetherian ring has only finitely many minimal prime ideals).

$\endgroup$
3
  • 1
    $\begingroup$ @Pierre-YvesGaillard Full disclosure: I have not thought this through, and it might be way off track.... but if I was going to try to come up with an example the first place I would look would be $R[x]$ where $R$ is a one-dimensional Pruefer domain with infinitely many primes and non-zero jacobson radical. e.g. appropriate localization of algebraic integers. I suspect you might find max (height 2) ideals containing infinitely many height one primes not contracting to $0$ in $R$, which would guarantee an element $f \in R[x]$ such that $R[x]/f$ has a max ideal as you desire. $\endgroup$ Jun 9 '20 at 15:01
  • 2
    $\begingroup$ Not exactly the same question, but your question is also answered by my example at math.stackexchange.com/questions/1542104/…. $\endgroup$ Jun 9 '20 at 16:50
  • $\begingroup$ @EricWofsey --- Thanks! If you posted a short answer linking to your other answer, I'd be glad to accept it. $\endgroup$ Jun 9 '20 at 18:17
5
$\begingroup$

Let $R$ be the quotient of a polynomial ring $k[x_1,x_2,\dots]$ in infinitely many variables over a field by the ideal generated by all products $x_ix_j$ for $i\neq j$. Geometrically, $\operatorname{Spec} R$ looks like infinitely many copies of $\mathbb{A}^1_k$ with their origins identified. So, $R$ is a $1$-dimensional ring with infinitely many minimal prime ideals, each of which are contained in the maximal ideal generated by all the $x_i$. See this answer for more details about this ring.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.