Hilbert's Basis Theorem
Zariski Topology
The Nullstellensatz: Algebra = Geometry!
Decomposing an Algebraic Set into Irreducible Components
Categorical Equivalence Part I (On Objects): Affine Algebraic Sets = Finitely Generated, Reduced k-Algebras
Categorical Equivalence Part II (On Morphisms): Morphisms of Affine Varieties = Morphisms of Finitely Generated, Reduced k-Algebras
Dimension Theory of Affine Varieties