# Ring spectrum

## From Wikipedia, the free encyclopedia

In stable homotopy theory, a **ring spectrum** is a spectrum *E* together with a multiplication map

*μ*:*E*∧*E*→*E*

and a unit map

*η*:*S*→*E*,

where *S* is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy, much in the same way as the multiplication of a ring is associative and unital. That is,

*μ*(id ∧*μ*) ∼*μ*(*μ*∧ id)

and

*μ*(id ∧*η*) ∼ id ∼*μ*(*η*∧ id).

Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava K-theory.