// Workers AI · dad joke modeWhat did the ring spectrum say? It had a resonating tone.

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 ∧ 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

• Adams, J. Frank (1974), Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-00523-2, MR 0402720