Intersections of ideals in $F[x]$

Let \(\{I_\lambda\}_{\lambda\in\Lambda}\) be any family of ideals in \(F[x]\) and put \[ I=\bigcap_{\lambda\in\Lambda}I_\lambda. \] Since every \(I_\lambda\) contains \(0\), so does \(I\). If \(f,g\in I\), then \(f,g\in I_\lambda\) for every \(\lambda\), hence \[ f-g\in I_\lambda \] for every \(\lambda\), so \(f-g\in I\). If \(h\in F[x]\) and \(f\in I\), then \[ hf\in I_\lambda \] for every \(\lambda\), so \(hf\in I\). Thus \(I\) is an ideal.