To assuage my conscience over an unsourced statement in a paper I'm writing:


I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or a statement of inexistence.



Maximal ideal of codimension >1
If the codimension is finite, then there is no such thing. If the codimension can be infinite, then yes, because there are infinite dimensional complex division algebras which are simple, like $\mathbb C(t)$.


