Why $P_1\neq P_1P_2$?

Question

Question: If $P_1,P_2$ are distinct prime ideals of an artinian ring, why is it that $P_1\neq P_1P_2$?

I know that prime ideals of an artinian ring are maximal, but still, I can't see why $P_1=P_1P_2$ is impossible. I suspect it must be very easy.

Answer 1

Note that $P_1P_2 \subseteq P_2$, so if $P_1 = P_1P_2$ then by maximality $P_1 = P_2$.

Answer 2

Note that you get $P_1=P_1P_2\subsetneq P_2$ , so $P_1$ is not maximal.