TITLE: Maximal Ideals in the Ring of Complex Entire Functions
QUESTION [8 upvotes]: Let $X = \mathcal{C}([0,1],\mathbb{R})$ be the ring of all continuous real-valued functions $f:[0,1] \to \mathbb{R}$. For $x \in [0,1]$, let $M_{x} = \\{ f \in M \ | \ f(x)=0\\}$. One can show by using compactness of $[0,1]$ that every maximal ideal is of this form.
Extending the Question to Entire functions: Let $\mathsf{C}(z)$ be the ring of complex entire functions. For $ \lambda \in \mathsf{C}$ let $M_{\lambda}$ denote the set of all entire functions which have a zero at $\lambda$. Then is $M_{\lambda}$ a maximal ideal in $\mathsf{C}(z)$, and does every ideal happen to be of this form? I don't know how to prove this!

REPLY [4 votes]: If I remember correctly from my student days, if D is a connected domain in the complex plane, then the ring O(D) of holomorphic functions on D has the property that every finitely generated ideal is actually principal, but there are ideas tat cannot be finitely generated (such as in Robin's answer above)