We say a commutative ring $R$ is a chain ring whenever its ideals form a chain with respect to inclusion. Is there a commutative chain ring $R$ that satisfies the following properties?
Property 1. $R$ has a non-finitely generated prime ideal $P$ with $P^2=0$.
Property 2. The unique maximal ideal, say $M$, of $R$ is principal and Krull dimension $R$ is one (the unique chain of prime ideals is $P\subsetneq M$).
Property 3. If $M=(x)$ then $x$ is regular (that is $ann (x)=0$) and $P=\bigcap_{i\in \Bbb N}M^i$.
Try $$R=k[[t]][s,s/t,s/t^2,\ldots]/(s^2,s^2/t^2,s^2/t^4,\ldots)$$
$R= k[[t]]\oplus k((t)) s$, $P=k((t)) s$, $M=(t)$, $(t)^k = t^k k[[t]]\oplus k((t)) s$.