Ring spectrum

Remarks

For a ring spectrum $E$, the coefficient group $\pi_*(E)$ is a graded ring. For a space $X$, the generalized cohomology theory $E^*(X)$ is a graded ring under the cup product induced by $\mu$ and the diagonal of $X$.

Example: for a commutative ring $R$, the Eilenberg–MacLane spectrum $HR$ is a commutative ring spectrum, with $\mu$ induced by the multiplication of $R$. It gives the usual cup product on $H^*(X; R)$. The sphere spectrum, complex K-theory $KU$ and complex cobordism $MU$ are other commutative ring spectra.

Non-example: $H(\Q/\Z)$ is not a ring spectrum. A unit and multiplication would make $\pi_0 = \Q/\Z$ a ring with identity $1$. If $1$ has additive order $n$, then $n x = (n \cdot 1) x = 0$ for all $x$, which fails for an element of order $2n$.

These axioms hold only up to homotopy. Ring spectra whose associativity and commutativity hold up to all higher coherent homotopies are called $A_\infty$ and $E_\infty$ ring spectra. They are modelled as monoids in symmetric spectra or orthogonal spectra.