Ring spectrum

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


 * μ: E &and; 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 &and; μ) ~ μ (μ &and; id)

and


 * μ (id &and; η) ~ id ~ μ(η &and; id).

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