On a special example of a Noetherian ring

Question

I am looking for an example of the following: A Commutative Noetherian ring $R$, an ideal $J$ of $R$ and an element $r\in R$ such that $r$ is not a zerodivisor on $R/J$ (i.e. $s\in R$ and $rs\in J $ implies $s\in J$ ) and $rJ\ne (r)\cap J$ .

Please help.

Answer 1

I think it’s impossible.

Let $R$ be a Noetherian ring, $J$ an ideal, and $r \in R$ be a regular element of $R/J$.

Then, for any $s \in R$, if $rs \in J$, then $s \in J$ and thus $rs \in rJ$. In other words $rJ \supset (r) \cap J$, so $rJ=(r) \cap J$.