Show that the ring $R$ is Noetherian

Question

Let $$R = \{ \frac{f(z)}{g(z)}: f, g \in \mathbb{C}[z], g(z) \neq 0 \text{ for } |z| = 1 \}.$$ Prove that $R$ is a Noetherian ring.

Note : $\mathbb C$ is the set of the complex numbers, $z$ is a complex variable, $f$ and $g$ are polynomials on the field of the complex numbers.

Answer 1

Hint. Show that $R$ is a ring of fractions of $\mathbb C[z]$.