---
tag: 6R57T5W7BV
kind: definition
name: Ring spectrum
msc: [55P43, 55P42]
defines: BAW0T68GWK
uses: [1GHF4EKBXS, 26GTNMAF7B, 14CGGHA85T, XJPYJT9Z1P, H28PSZR7XG]
sources:
  - url: https://en.wikipedia.org/wiki/Ring_spectrum
---

## Definition

Work in the [[stable homotopy category|26GTNMAF7B]], with smash product $\wedge$, unit the [[sphere spectrum|14CGGHA85T]] $S$, and twist isomorphism $\tau \colon E \wedge E \to E \wedge E$.
A *ring spectrum* is a [[spectrum|1GHF4EKBXS]] $E$ with morphisms
$$\mu \colon E \wedge E \to E, \qquad \eta \colon S \to E$$
such that:

1. (associativity) $\mu \circ (\mu \wedge \id_E) = \mu \circ (\id_E \wedge \mu)$ as morphisms $E \wedge E \wedge E \to E$;
2. (unit) $\mu \circ (\eta \wedge \id_E)$ and $\mu \circ (\id_E \wedge \eta)$ are the canonical isomorphisms $S \wedge E \to E$ and $E \wedge S \to E$.

The ring spectrum is *commutative* if moreover $\mu \circ \tau = \mu$.
In other words, a ring spectrum is a monoid in the stable homotopy category.

## Remarks

For a ring spectrum $E$, the coefficient group $\pi_*(E)$ is a graded ring.
For a space $X$, the [[generalized cohomology theory|H28PSZR7XG]] $E^*(X)$ is a graded ring under the cup product induced by $\mu$ and the diagonal of $X$.

Example: for a commutative ring $R$, the [[Eilenberg–MacLane spectrum|XJPYJT9Z1P]] $HR$ is a commutative ring spectrum, with $\mu$ induced by the multiplication of $R$.
It gives the usual cup product on $H^*(X; R)$.
The sphere spectrum, complex K-theory $KU$ and complex cobordism $MU$ are other commutative ring spectra.

Non-example: $H(\Q/\Z)$ is not a ring spectrum.
A unit and multiplication would make $\pi_0 = \Q/\Z$ a ring with identity $1$.
If $1$ has additive order $n$, then $n x = (n \cdot 1) x = 0$ for all $x$, which fails for an element of order $2n$.

These axioms hold only up to homotopy.
Ring spectra whose associativity and commutativity hold up to all higher coherent homotopies are called $A_\infty$ and $E_\infty$ ring spectra.
They are modelled as monoids in symmetric spectra or orthogonal spectra.