Let $\mathbb{F}[x]$ be a polynomial ring over a field $\mathbb{F}$. Let $I \subset \mathbb{F}[x]$ be an ideal, that is,
$\forall s,t \in I: s-t \in I$ and $\forall r \in \mathbb{F}[x]: rs,sr \in I$.
Prove that $\exists c(x) \in \mathbb{F}[x] : I = \{c(x)g(x): g(x) \in \mathbb{F}[x]\}$.
Any hints as to which $c(x) \in \mathbb{F}[x]$ I should pick and how I should proceed? Also my notes highlighted this result as particularly important... why is this? Thanks!