Affine varieties

Common polynomial zero sets define affine algebraic sets over a chosen field.

Different ways to understand it

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Coordinate rings
Polynomial functions on an affine algebraic set form a quotient by its vanishing ideal.
Localization
Localization formally inverts a chosen multiplicatively closed set of ring elements.
Polynomials
A polynomial is a finite sum of nonnegative integer powers with coefficients.
Projective space
Nonzero vectors modulo nonzero scalar multiplication represent projective points.
Rings
A ring supports addition and multiplication linked by distributive laws.