Let $F$ be a field. Prove that every ideal of the ring $F\left[X_{1}, \ldots, X_{n}\right]$ is finitely generated.

Consider the set

$$R=\left\{p(X, Y)=\sum c_{i j} X^{i} Y^{j} \in F[X, Y] \mid c_{0 j}=c_{j 0}=0 \text { whenever } j>0\right\}$$

Show that $R$ is a subring of $F[X, Y]$ which is not Noetherian.