Show that the polynomials in $\mathbb C[x,y]$ that contain no terms of the form $cy^m$ form a non-Noetherian subring

Question

I am trying to show that the polynomials in $\mathbb C[x,y]$ that contain no terms of the form $c y^m$ where $m>0$ and $c \in \mathbb C^*$ form a non-Noetherian subring.

I know that I need to find an ideal that isn't finitely generated but I am not sure how.

Answer 1

Hint: Show that $xy^{j+1}$ is not in the ideal containing $xy^{i}$ for $i\leq j$.