What is an example of proper ideal $M\subset \mathbb C[x,y]$ such that there is no homomorphism $\mathbb C[x,y]/M\to\mathbb C$?

Question

I am stuck in this problem for a while, and the main idea will be important for some exercises, so I really want to know how to find an example like this

I need an example of an proper ideal, let's say M, of C[x,y] (the ring of the complex polynomials in x and y) such that the quotient C[x,y]/M do not admit a homomorphism f(x,y) -> f(a, b) (for a, b complex numbers) from C[x,y]/M -> C.

Answer 1

Such an ideal does not exist. The maximal ideals of $\mathbb C[x, y]$ are all of the form $\mathfrak m = (x - a, y - b)$ for some $a, b \in \mathbb C$ (this is the weak Nullstellensatz). The quotient homomorphism $\mathbb C[x, y] \to \mathbb C[x, y]/\mathfrak m \simeq \mathbb C$ is simply the evaluation map $f(x, y) \mapsto f(a, b)$.

Now if $M$ is any proper ideal then $M$ is contained in some maximal ideal $\mathfrak m = (x - a, y - b)$. The natural homomorphism $\mathbb C[x, y]/M \to \mathbb C[x, y]/\mathfrak m \simeq \mathbb C$ is of the form $f(x, y) \mapsto f(a, b)$.