Proposition 15

In a ring $A$ let $a$ be an ideal and $b$ a finitely generated ideal contained in the radical of $a$. Then there exists an integer $k > 0$ such that $b^k \subset a$.

Let $(b_i)_{1 \leq i \leq n}$ be a system of generators of $b$. By hypothesis there exists an integer $h$ such that $b_i^h \in a$ for $1 \leq i \leq n$. If a product of $nh$ elements, each of which is a linear combination of the $b_i$ with coefficients in $A$, is expanded, each term is a multiple of a product of $nh$ factors, each of which is equal to a $b_i$; among these factors, at least $h$ have the same index $i$ for some $i$, hence, the product belongs to $a$ and $nh$ is the required integer $k$.