Projective space

Nonzero vectors modulo nonzero scalar multiplication represent projective points.

Different ways to understand it

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Affine varieties
Common polynomial zero sets define affine algebraic sets over a chosen field.
Morphisms of varieties
Regular maps preserve algebraic structure through locally appropriate polynomial or rational expressions.
Algebraic dimension
Chains of prime ideals define Krull dimension and connect algebraic structure to geometric dimension.
Algebraic singularities
Singular points fail suitable smoothness conditions measured by local geometry or tangent behavior.
Coordinate rings
Polynomial functions on an affine algebraic set form a quotient by its vanishing ideal.