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.
References
Ring spectrum Wikipedia(Text) CC BY-SA