I want to lead this sequence to the point where I am having trouble understanding algebraic geometry. Hence, I won't take the usual course such an introduction would take, but rather set the stage reasonably quickly to make the transit to the more abstract themes clear.
But that's all a few posts away. For now, recall that we recognized already that any variety is defined by an ideal, and that intersections and unions of varieties are given by sums and intersections or products of ideals.
This is the first page of what is known as the Algebra-Geometry dictionary. The dictionary is made complete by a pair of reasonably famous theorems. I won't bother proving them - the proofs are a good chunk of any decent commutative algebra course - but I'll quote the theorems and discuss why they matter.
We call a ring Noetherian if all ideals are finitely generated. If a ring R is Noetherian, then quotients are Noetherian.
Hilbert's Basis Theorem: If R is Noetherian, then so is R[x].
We define the radical [tex]\sqrt I[/tex] of an ideal I to be the ideal consisting of all elements a such that some power of a is actually in I. We call an ideal I radical if [tex]I=\sqrt I[/tex]. This concept is relevant for our considerations since if for a point p the function f^n(p) vanishes, then f(p) also vanishes. Thus, the set of points such that f^n vanishes is the same set as the set of points where f vanishes. The relevancy of this is captured in:
Note, for the statement of this theorem that we write V(I) for the variety defined by simultaneous vanishing of all elements in I, and we write I(V) for the ideal of all polynomials in [tex]k[x_1,\dots,x_n][/tex] that vanish on all of I(V).
So - and here is the beautiful part - affine algebraic varieties correspond bijectively to radical ideals in polynomial rings. For every ideal, there is a variety and for every variety, there is an ideal. But we can push this further.
Let's consider polynomial functions from [tex]\mathbb A^n[/tex] to k. These are precisely the polynomials in [tex]k[x_1,\dots,x_n][/tex]. Given a variety V, we can take a polynomial [tex]f\in k[x_1,\dots,x_n][/tex] and restrict it to a function [tex]f|_V:V\to k[/tex].
Two different polynomials give the same restricted function precisely when their difference vanishes on all of V. So polynomial functions on V are precisely the equivalence classes in the quotient ring [tex]k[x_1,\dots,x_n]/I(V)[/tex]. We call the resulting ring the coordinate ring and denote it by k[V].
Conversely, if R is a Noetherian k-algebra such that there are no nilpotent elements in R, then R is a quotient of some polynomial ring with some radical ideal. Hence it is the coordinate ring of some variety in some affine space somewhere. We call a ring lacking nilpotents reduced.
The really beautiful part will come in my next post. We can introduce homomorphisms of varieties in a reasonably natural way so that this bijective correspondence ends up being functorial - i.e. any homomorphisms on one side gives rise to a corresponding homomorphism on the other side. Thus, the categories of Noetherian reduced k-algebras and of affine algebraic varieties are equivalent.