---
tag: FB45BHHA2J
kind: concept
name: Spectrum of a ring
msc: [14A05, 13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_of_a_ring
same_as:
  wikidata: Q1154351
  wikipedia: Spectrum of a ring
---

## Summary

The spectrum $\operatorname{Spec} R$ of a [[commutative ring|0310REJJY8]] $R$ is the set of its [[prime ideals|00MHHZ9SNE]], with the [[Zariski topology|8SQK7RTB63]].
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.

## Explanations

### L1 human

For the integers, the points of $\operatorname{Spec} \Z$ are the ideals $(2), (3), (5), \ldots$, one for each prime number, and one extra point $(0)$.
An integer $n$ "vanishes" at the point $(p)$ when $p$ divides $n$, so a nonzero integer $n$ behaves like a function with finitely many zeros.
For the polynomial ring $\C[x]$, the maximal ideals $(x - a)$ correspond to the complex numbers $a$, so the spectrum contains a copy of the complex line.
The [[definition|QJAXM2SF1Z]] gives the set and its topology.