So in algebraic geometry, the prime spectrum of the ring of polynomials over a field k is isomorphic to the set of k-homomorphisms from the ring of polynomials to k, and we do this by using the fact that the residue field (localise w.r.t. some prime ideal, then mod out the ideal generated by this in the localisation) is isomorphic to k, and hence any polynomial goes to an element of k, via the equivalence relations. This all makes sense as the ideal generated by a prime ideal of the polynomial ring in its localisation is clearly maximal, but I'm stuck on a very basic example:
Let C[x,y] be the polynomial ring over the complex numbers and (x - 1) be the prime ideal. What number does y go to? A C-algebra homomorphism is defined completely by its action on the generators of the polynomial ring (so just the n variables, and taking n = 2 we get x |--> 1, but y |--> ?), but I can't figure out how this works in this simple case. It should maybe be zero (otherwise how do you invert 1 + y?), but I don't know. Can anyone explain this?