The irreducible components of Spec(A)

Question

Today when I read Commutative algebra By Atiyah, after I proved that closed set $V(\mathfrak{P})$, where $(\mathfrak{P})$ is a minimal prime ideal of $A$, is irreducible subspace. But I meet a problem on how to prove $V(\mathfrak{P})$ is the maximal irreducible subspace. Can anyone help me?

Answer 1

Suppose that $p$ is a minimal prime and there exists a prime $q$ such that $V(p)\subset V(q)$, this implies that $p\in V(q)$ and $q\subset p$, we deduce that $p=q$ since $p$ is a minimal prime.