Spectrum of a ring

Summary

The spectrum $\operatorname{Spec} R$ of a commutative ring $R$ is the set of its prime ideals, with the Zariski topology. It turns a ring into a geometric space, on which the elements of $R$ behave like functions. A ring homomorphism $R \to A$ gives a continuous map $\operatorname{Spec} A \to \operatorname{Spec} R$ in the opposite direction. The spectrum is the underlying space of an affine scheme, the basic object of modern algebraic geometry.