= Solution
In the <stable homotopy category>, a <ring spectrum> is a unital associative monoid object for the <smash product of spectra>. Thus it consists of a <topological spectrum> $E$ with multiplication and unit
$$
\mu:E\wedge E\longrightarrow E,\qquad\eta:\mathbb S\longrightarrow E,
$$
where $\mathbb S$ is the <sphere spectrum>, and the identities are
$$
\mu(\mu\wedge1)=\mu(1\wedge\mu),\qquad\mu(\eta\wedge1)=1_E=\mu(1\wedge\eta).
$$
The canonical associativity and unit identifications of the <symmetric monoidal category> are understood in these formulas. All identities are identities of maps in the <stable homotopy category>.
A <commutative ring spectrum> also satisfies
$$
\boxed{\mu\tau=\mu,}
$$
where $\tau:E\wedge E\to E\wedge E$ interchanges the two factors. This is the ring-object convention; a coherently structured ring spectrum has additional point-set or higher-homotopy data. In particular homotopy-category commutativity alone is not an assertion of such a coherent enhancement.
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