Spectrum of a ring

Prime ideals form a topological space whose basic structure encodes a commutative ring.

Different ways to understand it

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Prime ideals
For a prime ideal in a commutative ring, a product inside the ideal forces a factor inside.
Schemes
Schemes glue locally ringed spaces modeled on spectra of commutative rings.
Affine varieties
Common polynomial zero sets define affine algebraic sets over a chosen field.
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.