Ring spectrum

For the concept of spectrum of a ring in algebraic geometry, see spectrum of a ring.

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

μ:E EE

and a unit map


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)


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.

See also


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