# Essential Prime Ideal

I search for an example of a commutative ring $R$ with unity having a prime ideal $P$ and some element $r\in R$ such that the annihilator of $r$ is both contained in $P$ and essential in $R$. By being essential I mean that the intersection of the annihilator with any nonzero ideal of $R$ is nonzero.

Since this (essential) annihilator is a subset of the prime ideal $P$, we infer that $P$ would be essential in $R$ too.

Thanks for any help or suggestion!

In $\Bbb Z/4\Bbb Z$, the annihilator of $2+4\Bbb Z$ is $(2+4\Bbb Z)$ which is the unique maximal (and unique prime, and unique nontrivial, for that matter) ideal of the ring. (Incidentally, this annihilator is also a superfluous ideal).
• Thanks for the answer, but in fact, I want $P$ to be, of course, non-maximal. Does there exist any example? Thanks again! – karparvar Jun 4 '16 at 6:10