Proposition 1

Let $\mathbf{A}$ be a ring, $a_1, a_2, \ldots, a_n$ ideals of $\mathbf{A}$ and $p$ a prime ideal of $\mathbf{A}$. If $p$ contains the product $a_1 a_2 \ldots a_n$, it contains at least one of the $a_i$.

Suppose in fact that $p$ contains none of the $a_i$. For $1 \leq i \leq n$ there exists then an element $s_i \in a_i \cap p$; then $s = s_1 s_2 \ldots s_n$ is contained in $a_1 a_2 \ldots a_n$, and is not contained in $p$, which is absurd.