Let $R:=C(\mathbb R,\mathbb R)$ be the ring of continuous $\mathbb R$-valued functions on $\mathbb R$. Is every prime ideal in $R$ maximal?
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1This is rather catastrophically false. See https://math.stackexchange.com/questions/2078755/finding-a-space-x-such-that-dim-cx-n for a general result which shows how badly this fails. – Eric Wofsey Sep 27 '18 at 05:23
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@EricWofsey Thank you for pointing that out. – principal-ideal-domain Sep 27 '18 at 18:58
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No, because that would imply $J(R)$ is nil and $R/J(R)$ is von Neumann regular. (I am denoting the Jacobson radical of $R$ as $J(R)$.)
But it is easy to show in this ring that $J(R)$ is $\{0\}$ and $R$ is not von Neumann regular. (In fact, this ring has only trivial idempotents, but there must be many idempotents in a von Neumann regular ring.)
You can read elsewhere about the connection of prime ideals in this ring to ultrafilters. This is pretty much the best description you can get for prime ideals in such a ring, and it shows that in general prime ideals in a ring like this do not have to be maximal.
rschwieb
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