---
tag: BAW0T68GWK
kind: concept
name: Ring spectrum
msc: [55P43, 55P42]
specializes: [1GHF4EKBXS]
sources:
  - url: https://en.wikipedia.org/wiki/Ring_spectrum
same_as:
  wikidata: Q7334821
  wikipedia: Ring spectrum
---

## Summary

A ring spectrum is a [[spectrum|1GHF4EKBXS]] with a multiplication and a unit that satisfy the ring axioms up to homotopy.
It is the stable analogue of a ring, with the [[sphere spectrum|14CGGHA85T]] in the role of the integers.
The cohomology theory represented by a ring spectrum has cup products, so its values on spaces are graded rings.

## Explanations

### L1 human

A ring is a set with a multiplication that has a unit and is associative.
A ring spectrum is the same idea one level up: a spectrum $E$ with a map $E \wedge E \to E$ and a unit map from the sphere spectrum.
Ordinary cohomology with coefficients in a ring, complex K-theory and complex cobordism all come from ring spectra.
This is why in each of them one can multiply two cohomology classes.