# nLab E-infinity-ring

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

An ${E}_{\infty }$-ring is a commutative monoid in the stable (∞,1)-category of spectra. Sometimes this is called a commutative ring spectrum. An E-∞ algebra in spectra.

This means that an ${E}_{\infty }$-ring is an A-∞ ring that is commutative up to coherent higher homotopies. ${E}_{\infty }$-rings are the analogue in higher algebra of the commutative rings in ordinary algebra.

In terms of model categories, and ${E}_{\infty }$-rings may be modeled as ordinary commutative monoids with respect to the symmetric monoidal smash product of spectra, a fact sometimes referred to as “brave new algebra”.

## Examples

E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
• Peter May with contributions by Frank Quinn, Nigel Ray and Jorgen Tornehave, ${E}_{\infty }$-Ring spaces and ${E}_{\infty }$ ring spectra (pdf)