Prime ideals

For a prime ideal in a commutative ring, a product inside the ideal forces a factor inside.

Different ways to understand it

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Localization
Localization formally inverts a chosen multiplicatively closed set of ring elements.
Spectrum of a ring
Prime ideals form a topological space whose basic structure encodes a commutative ring.
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.