Is there an example of commutative ring with exactly three prime ideals for which this property holds? 
Is there an example of commutative ring with exactly three non zero prime ideals $P_i$ which satisfies the following statement: $P_1P_2=0$ and for an ideal $I\neq 0$ such that $I\neq P_i$ we have $IP_3=0$.

Note. On January 15, 2014 this user posted and later removed the above question (which I don't remember if it was answered or not).

So this turned out to be much easier than it seems. Take $R = k[[x,y,z]]/(x,y)\cdot(x,z)$. This ring has exactly $3$ primes, namely $(x,y,z), (x,y), (x,z)$, and the maximal ideal $(x,y,z)$ is killed by the nonzero element $x$, while the two minimal primes annihilate each other.
