Proposition

Let $I \subset k[x_1, \cdots, x_n]$ be an ideal, and let $f_1, \cdots, f_s \in k[x_1, \cdots, x_n]$. Prove that the following statements are equivalent:

$f_1, \cdots, f_s \in I$.
$\ev{ f_1, \cdots, f_s } \subset I$.

Solution

Since $I$ is closed under addition and closed under multiplication by elements in $k[x_1, \cdots, x_n]$, if $1$ is true, $\sum h_if_i \in I$ for any $h_i$’s. Thus $1 \implies 2$.

Since $f_i \in \ev{ f_1, \cdots, f_s }$ for each $i$, $2 \implies 1$.