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I'm asked to show that the ring of holomorphic functions on the unit disc $\{z \in \mathbb{C} \mid |z| < 1\}$ is not a local ring.

I'm quite sure that this is not a difficult proof, and I've already done some work with this ring, showing in particular that it is an integral domain, but for some reason I'm struggling with this particular fact. Hints would be greatly appreciated.

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  • $\begingroup$ Find two different maximal ideals. Hint: there are lots of surjective homomorphisms to $\mathbb{C}$... $\endgroup$ – Zhen Lin Sep 19 '13 at 17:02
  • $\begingroup$ Function vanishing at a point $z_0$ form a maximal ideal, so... $\endgroup$ – user8268 Sep 19 '13 at 17:03
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Hint:

Show that $\{f\mid f(1/2)=0\}$ and $\{f\mid f(-1/2)=0\}$ form distinct maximal ideals.

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    $\begingroup$ This should also give you a pretty good hint as to why they are called "local rings"! $\endgroup$ – Steven Gubkin Sep 19 '13 at 17:28
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Alternatively, you can do this without even having to try and produce maximal ideals. A ring $R$ is a local if and only if the non-units $R-R^\times$ form an ideal.

Note that $z$ and $\frac{1}{2}-z$ are both non-units (the first has a zero at $0$ and the second at $\frac{-1}{2}$). But, their sum is $\frac{1}{2}$ which is evidently a unit.

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