title: Ring spectrum
text: # Ring spectrum
  
  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.
  
  