# Maximal ideal of codimension >1

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.

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 $\mathbb C(t)$.
• Miguel, if a maximal ideal $\mathfrak m$ has finite codimension in your $\mathbb C$-algebra $A$, then $A/\mathfrak m$ is a finite dimensional commutative $\mathbb C$-algebra which is simple. It is therefore a finite dimensional division $\mathbb C$-algebra, and it must then be of dimension $1$, as it is in fact an agebraic field extension of $\mathbb C$. Apr 12, 2010 at 19:19
By codimension you just mean as a $\mathbb{C}$-vector space? Take the rational function field $\mathbb{C}(t)$.
(Note: by the Nullstellensatz, it is not possible to do so with a finitely generated $\mathbb{C}$-algebra.)