Prime ideal being maximal ideal in an affine algebra

Question

Let $A$ be an affine algebra over a field and $P$ be a prime ideal of $A$. If $P$ is a finite intersection of maximal ideals, then $P$ is maximal.

Is this statement true? And if so, how to prove it?

Any suggestions will be appreciated, thanks!

Answer 1

This is true in any commutative ring. Even more generally, suppose $A$ is a commutative ring and $I_1,\dots,I_n,P\subseteq A$ are ideals such that $P$ is prime and $I_1\cap \dots \cap I_n\subseteq P$. Then $I_k\subseteq P$ for some $k$ (in particular if $I_k$ is maximal this means $I_k=P$ so $P$ is maximal).

To prove this, just note that $I_1\dots I_n\subseteq I_1\cap \dots \cap I_n\subseteq P$, and since $P$ is prime if it contains a product of ideals it must contain one of the factors.