I'm looking for a textbook reference of the following elementary fact (a reference for an excercise in a textbook is also welcome):

Let $R$ be a commutative ring and let $\mathfrak{p}$ be a prime ideal of $R$. An ideal $P \trianglelefteq R[X]$ with $P \cap R = \mathfrak{p}$ is prime iff $P = \mathfrak{p}R[X]$ or if $P$ is maximal in $\lbrace I \trianglelefteq R[X] \mid I \cap R = \mathfrak{p} \rbrace$.