Affine Varieties and the Vanishing Ideal

Can someone please explain to me these two concepts and how they differ from each other? I'm taking a class entitled Math & Computers which emphasizes algebraic geometry in a symbolic-computational setting. That being said, the class is not very oriented towards explaining and understanding these ideas as it is assumed that I've already had experience in dealing with them.

Most specifically, I'm talking about affine varieties and vanishing ideals in the realm of polynomial rings. Is the vanishing ideal a subset of the affine variety?
 
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If you have a subset Z of your affine space (for example Rn), the corresponding vanishing ideal, is the set of polyomials in n variables being zero on every point of Z. If you have a set T of polynomials, you can talk about the common zeros of all polynomials in T, which will be a subset of Rn, and call this an affine variety, usually with some irreducibility-condition, depending on your text-book.

So loosely speaking, your objects, varieties and ideals, live in two different worlds, Rn and the polynomial ring, respectively. But a variety gives an ideal, and an ideal gives a variety. To be more precise, you can prove a theorem showing a one-to-one correspondence between the (irreducible) affine varieties in Rn, and prime ideals in the polynomial ring, and by using this theorem you can translate statements about ideals to statements about varieties, and vice versa.
 
I had a moment of insight last night around 1am. I understand now how ideals and affine varieties are linked. The vanishing ideal is the ideal of a variety. Ideals also give us a way to compute with affine varieties, correct?
 

mathwonk

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varieties consist of points where certain polynomials, the ones in the vanishing ideal, vanish.

"vanish" means equal zero.
 

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