I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann\_R(r)\in {\rm Min}(R)$ we have $Ann\_R(P)=Rr$.


This question is a follow-up of my previous [question](https://mathoverflow.net/questions/129949/when-does-second-annihilator-of-a-principal-ideal-equal-the-ideal-itself-ie)



An example of a ring $R$  with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.
I believe the graded ring $k[x,y]/(xy)$ satisfies this property. The minimal primes are just $(x)$, $(y)$ and the annihilators are just the other ideal. The dimension is 1. This is not however local.


