Show that a ring $R$ is Noetherian if and only if every ideal of $R$ is finitely generated. Show that if $\phi: R \rightarrow S$ is a surjective ring homomorphism and $R$ is Noetherian, then $S$ is Noetherian.

State and prove Hilbert's Basis Theorem.

Let $\alpha \in \mathbb{C}$. Is $\mathbb{Z}[\alpha]$ Noetherian? Justify your answer.

Give, with proof, an example of a Unique Factorization Domain that is not Noetherian.

Let $R$ be the ring of continuous functions $\mathbb{R} \rightarrow \mathbb{R}$. Is $R$ Noetherian? Justify your answer.