If X is a subset of C^n, define Fun(X) := C[x_1..x_n] / I(X).
Def. ring homomorphism and isomorphism.
Example. If X = {xy=0}, so I(X) = < xy >, then Fun(X) is not isomorphic to C[z].
Proof: Fun(X) has "zero divisors", and C[z] doesn't.
Lemma. If p,q are polynomials in C[z], q nonzero, then there exist polynomials m,r such that p = mq + r, and deg(r) < deg(q). (If r is 0, we define its degree to be negative infinity.)
Theorem. Any ideal in C[z] is principal (generated by one element).
