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If $E$ is a ring spectrum, then $F(X_{+},E)$ is a ring spectrum. Its multiplication is adjoint to evaluating the two functions on the two copies supplied by the diagonal of $X$, then multiplying in $E$. Its unit is the constant unit function. Coassociativity of the diagonal and associativity in $E$ prove the associative identity; the counit and unit identities prove both unit laws.

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 unitwhere $\mathbb{S}$ is the sphere spectrum, and the identities areThe 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 satisfieswhere $\tau :E\wedge E\to E\wedge E$ 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.

Put $A={\Sigma }^{\infty }{X}_{+}$ and $H=F(A,E)=F(X_{+},E)$. The diagonal of $X$ and its map to a point induceHere the identification $(X\times X{)}_{+}\cong X_{+}\wedge X_{+}$ explains the target of $\Delta$. These maps satisfy the coassociativity, cocommutativity and counit identities, since their underlying maps send $x$ to repeated copies of the same $x$.

Use the defining adjunction of the function spectrum to define $m:H\wedge H\to H$. Its adjoint is the compositeDefine $u:\mathbb{S}\to H$ to have adjointThus $m$ is pointwise multiplication, while $u$ is the constant unit. These are genuine stable maps, defined through the internal function spectrum, so the argument does not presume that stable functions have ordinary points.

To verify associativity, take the adjoints of $m(m\wedge 1)$ and $m(1\wedge m)$. Both are maps $H^{\wedge 3}\wedge A\to E$. Both use the triple diagonal $A\to A^{\wedge 3}$, evaluate the three copies of $H$, and multiply the three outputs. The triple diagonals agree by coassociativity; the output maps agree by associativity of the ring spectrum $E$. The function spectrum adjunction therefore makes the original two maps equal. For the left unit, the adjunct of $m(u\wedge 1)$ uses $(ϵ\wedge 1)\Delta =1_A$ and $\mu (\eta \wedge 1)=1_E$, leaving exactly evaluation. Its adjoint is $1_H$. The same calculation with $(1\wedge ϵ)\Delta$ proves the right unit. HenceIf $E$ is a commutative ring spectrum, cocommutativity of $\Delta$ proves that this function spectrum is commutative as well.

A ring spectrum is a unital associative monoid object of the stable homotopy category under the smash product of spectra. It has $\mu :E\wedge E\to E$ and $\eta :\mathbb{S}\to E$, satisfying the associative and two unit identities. Structured ring spectra additionally specify coherent point-set or higher homotopies; a ring object alone does not assert those enhancements.