TITLE: The ring of convergent power series over $\mathbb C$ isn't noetherian
QUESTION [5 upvotes]: How can one prove that the ring of convergent everywhere power series in $\mathbb C[[z]]$ isn't Noetherian?

REPLY [5 votes]: Alternatively, in every Noetherian ring, any non-unit, non-zero element is, up to units, a finite product of irreducibles. The irreducibles in $\mathcal{O}(\mathbb{C})$ are easily seen to be the functions $z-a$. Thus, if $\mathcal{O}(\mathbb{C})$ was Noetherian, then every non-unit, non-zero element would need to be a polynomial. This is certainly not true :)