Localization

Localization formally inverts a chosen multiplicatively closed set of ring elements.

Different ways to understand it

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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.
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.
Algebraic singularities
Singular points fail suitable smoothness conditions measured by local geometry or tangent behavior.