I am trying to prove that the Nilradical of a ring $A$ is the intersection of the minimal prime ideals. The one inclusion is obvious but I cannot do the other. I am just taking an $x$ which belongs to the intersection of all minimal prime ideals and I am trying to prove that it belongs to the intersection of all primes.
I know that a prime ideal $p$ is minimal if there is no prime between the $(0)$ and $p$. But how can I relate an arbitrary prime ideal with the minimal ones? Does it hold that every prime ideal contains a minimal prime ideal over the $(0)$ ideal?
