= Solution
We use the <James reduced product> theorem and the <Bott–Samelson theorem>: if $X$ is a connected based <CW complex> with free integral <homology>, then $JX\to\Omega\Sigma X$ is a weak equivalence, and its induced <Pontryagin ring> is
$$
H_*(\Omega\Sigma X;\mathbb Z)
\cong T_{\mathbb Z}\bigl(\widetilde H_*(X;\mathbb Z)\bigr).
$$
Here $\Sigma$ is the <reduced suspension>, $T$ is the <tensor algebra>, and the multiplication is induced by concatenating James words, hence by concatenating loops. Taking $X=S^{2n}$ supplies a single generator $x$ of degree $2n$, so
$$
H_*(\Omega S^{2n+1};\mathbb Z)=\mathbb Z\{1,x,x^2,\ldots\}
$$
is free, with one generator in every degree $2nk$ and zero in the other degrees.
The diagonal makes this a <homology> coalgebra. The generator $x$ is a <primitive homology class>:
$$
\Delta x=x\otimes1+1\otimes x.
$$
There are no nontrivial lower positive degrees in which its reduced diagonal could land. Compatibility of the diagonal with loop multiplication gives
$$
\Delta(x^k)=(x\otimes1+1\otimes x)^k
=\sum_{i=0}^k\binom ki x^i\otimes x^{k-i}.
$$
The degree of $x$ is even, so the two tensor factors commute without a <Koszul sign rule>.
The <universal coefficient theorem for cohomology> has no Ext terms here because <homology> is free. Let $a_k$ be the <cohomology> class dual to $x^k$, with $a_0=1$ and $|a_k|=2nk$. The <cup product> is dual to the diagonal, hence
$$
\boxed{a_i a_j=\binom{i+j}{i}a_{i+j}.}
$$
Therefore \b[the integral <cohomology> ring is the <divided power algebra>]
$$
\boxed{H^*(\Omega S^{2n+1};\mathbb Z)
\cong\Gamma_{\mathbb Z}(a),\qquad |a|=2n.}
$$
Explicitly $H^{2nk}=\mathbb Z a_k$ for $k\geq0$, and all other <cohomology> groups vanish. In particular $a_1^k=k!a_k$: over the integers this is not a <polynomial ring> on $a_1$. This is the <integral cohomology of an odd-sphere loop space>.
For the requested <homology> multiplication, use the <homology cross product> followed by concatenation:
$$
\boxed{\alpha*\beta=\mu_*(\alpha\times\beta).}
$$
The constant loop gives the degree-zero unit. Loop concatenation is associative up to <homotopy>, which suffices for associativity on <homology>; Moore loops can make the space-level operation strictly associative. The <Künneth theorem> identifies the tensor-product <homology> because its groups are <free abelian groups>. The <Bott–Samelson theorem> identifies this product with word multiplication, so $x^i*x^j=x^{i+j}$. \b[The <Pontryagin ring of an odd-sphere loop space> has presentation]
$$
\boxed{H_*(\Omega S^{2n+1};\mathbb Z),*
\cong T_{\mathbb Z}(x)\cong\mathbb Z[x],\qquad |x|=2n.}
$$
With only one generator the <tensor algebra> has the indicated polynomial presentation. The <homology> product and the divided-power <cohomology> product are different operations.
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