Why does Proposition 1.8 in Atiyah-Macdonald imply that the smallest prime $\mathfrak{p}$ containing a primary ideal is equal to its radical?

Question

Proposition 4.1 in Atiyah-Macdonal states that the radical of a primary ideal is the smallest prime ideal containing the primary ideal. They start the proof claiming that showing the radical is a prime ideal is enough because of proposition 1.8. This proposition gives us that the nilradical is equal to the intersection of all prime ideals. I don't see how proposition 1.8 gives us this. Are there any claims I could prove which would make this statement more obvious?

Answer 1

Proposition 1.8 says that $r(\mathfrak{q})$ is the intersection of all the prime ideals containing $\mathfrak{q}$. If $r(\mathfrak{q})$ is itself a prime ideal, then in fact

• $r(\mathfrak{q})$ is a prime ideal containing $\mathfrak{q}$ (the intersection of sets containing $\mathfrak{q}$ will itself contain $\mathfrak{q}$)
• $r(\mathfrak{q})$ is contained within all other prime ideals containing $\mathfrak{q}$ (being the intersection of them)

which is the definition of being the smallest prime ideal containing $\mathfrak{q}$.