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).
What you are looking for is any commutative ring which is not nonsingular. You can pick any nonzero element of the singular ideal, and its ideal will be essential. It will always be contained in some maximal (hence prime) ideal, so that will give you your prime.
Here is a list of some more examples.