2
$\begingroup$

Does there exist a $1$-dimensional Noetherian domain whose set of all prime ideals has cardinality $\aleph_1$ ?

What I know is that for $1$-dimensional Noetherian domain $R$, $\operatorname{max} \operatorname {Spec} R$, under Zariski subspace topology of $\operatorname{Spec} R$, is actually the co-finite topology. But I don't think that would help implying existence of any such domain with specified cardinality of prime spectrum.

$\endgroup$
3
$\begingroup$

Let $K$ be an algebraically closed field. Then the polynomial ring $K[X]$ has $|K|$ prime ideals, namely $\{0\}$ and the $\left<X-a\right>$ for $a\in K$. There is an algebraically closed field of each infinite cardinality.

| cite | improve this answer | |
$\endgroup$

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.