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  "body": "\n## Definition\n\nWork in the [[stable homotopy category|26GTNMAF7B]], with smash product $\\wedge$, unit the [[sphere spectrum|14CGGHA85T]] $S$, and twist isomorphism $\\tau \\colon E \\wedge E \\to E \\wedge E$.\nA *ring spectrum* is a [[spectrum|1GHF4EKBXS]] $E$ with morphisms\n$$\\mu \\colon E \\wedge E \\to E, \\qquad \\eta \\colon S \\to E$$\nsuch that:\n\n1. (associativity) $\\mu \\circ (\\mu \\wedge \\id_E) = \\mu \\circ (\\id_E \\wedge \\mu)$ as morphisms $E \\wedge E \\wedge E \\to E$;\n2. (unit) $\\mu \\circ (\\eta \\wedge \\id_E)$ and $\\mu \\circ (\\id_E \\wedge \\eta)$ are the canonical isomorphisms $S \\wedge E \\to E$ and $E \\wedge S \\to E$.\n\nThe ring spectrum is *commutative* if moreover $\\mu \\circ \\tau = \\mu$.\nIn other words, a ring spectrum is a monoid in the stable homotopy category.\n\n## Remarks\n\nFor a ring spectrum $E$, the coefficient group $\\pi_*(E)$ is a graded ring.\nFor a space $X$, the [[generalized cohomology theory|H28PSZR7XG]] $E^*(X)$ is a graded ring under the cup product induced by $\\mu$ and the diagonal of $X$.\n\nExample: for a commutative ring $R$, the [[Eilenberg–MacLane spectrum|XJPYJT9Z1P]] $HR$ is a commutative ring spectrum, with $\\mu$ induced by the multiplication of $R$.\nIt gives the usual cup product on $H^*(X; R)$.\nThe sphere spectrum, complex K-theory $KU$ and complex cobordism $MU$ are other commutative ring spectra.\n\nNon-example: $H(\\Q/\\Z)$ is not a ring spectrum.\nA unit and multiplication would make $\\pi_0 = \\Q/\\Z$ a ring with identity $1$.\nIf $1$ has additive order $n$, then $n x = (n \\cdot 1) x = 0$ for all $x$, which fails for an element of order $2n$.\n\nThese axioms hold only up to homotopy.\nRing spectra whose associativity and commutativity hold up to all higher coherent homotopies are called $A_\\infty$ and $E_\\infty$ ring spectra.\nThey are modelled as monoids in symmetric spectra or orthogonal spectra.\n",
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