Let $R$ be a Noetherian local ring of dimension $\geq 3$ and $\{p_1,\ldots , p_n\}$ be a collection of prime ideals of height $1$. Does there exist an element $x\in R$ such that $Ass(R/xR)=\{p_1,\ldots , p_n\}$.
$\begingroup$
$\endgroup$
3
-
$\begingroup$ No. Try the case of a Dedekind domain and $n=1$. $\endgroup$– MohanCommented Mar 14, 2023 at 17:12
-
$\begingroup$ @Mohan I have edited the question. Can we say anything if R is a domain or R has no embedded associated primes with dimension>=3? $\endgroup$– CuspCommented Mar 14, 2023 at 18:06
-
$\begingroup$ If you have an example in dimension $n$, taking polynomial ring over it, you get an example in dimension $n+1$. $\endgroup$– MohanCommented Mar 15, 2023 at 2:22
Add a comment
|