{{For|the concept of spectrum of a ring in algebraic geometry|spectrum of a ring}}
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.
== See also == *Highly structured ring spectrum
==References== {{reflist}} *{{citation|mr=402720 |last=Adams|first= J. Frank|isbn=0-226-00523-2 |title=Stable homotopy and generalised homology |publisher=University of Chicago Press|series = Chicago Lectures in Mathematics|year=1974}}
Category:Algebraic topology Category:Spectra (topology) de:Ringspektrum
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