We defined polynomial rings (with complex coefficients), and ideals therein.
Monomial ideals.
Two natural ways to produce ideals:
1) Any subset X of C^n has an I_X of those polynomials that vanish at every point of X.
Example: if X is the y-axis in C^2, then I_X in C[x,y] is all multiples of x.
2) Given any set in a polynomial ring R, we can generate an ideal from it by taking all R-linear combinations.
Big theorem to come: any ideal has a finite generating set.
(Warning: in the links above, the ring R may not be a polynomial ring, and may not even be commutative. That leads to a bunch of extra worries, like "left ideals" vs. "right ideals", that are not our problem.)
