[Math] When is nilradical not a prime ideal

Question

Atiyah gives this criterion for nilradical to be a prime ideal.Nilradical is the intersection of prime ideals.Is nilradical prime iff there is only one prime ideal? ie Intersection of distinct prime ideals can never be a prime ideal?

Answer 1

Nilradical is intersection of prime ideals. So if there is more than one minimal prime ideal, then nilradical is not a prime ideal.
note that Intersection of distinct minimal prime ideals can never be a prime ideal because if $p_1\cap... \cap p_n = p$ then $p_1... p_n \subset p$ so $p_i \subset p$ for some i. this means $p=p_i$