Ring spectrum

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

μ: E EE

and a unit map

η: SE,

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.