I have a CW complex $K$ and a loop space $\Omega K$. By $Q(\Omega K)$ I denote the group generated by the singular cubes of $\Omega K$. Then the book says that "the multiplication in $\Omega K$ induces ring structure in $Q(\Omega K)$ in the usual way". What is this usual way? All I can think of is that for $f,g:[0,1]^n\to \Omega K$ their product $h=fg$ works like that: $$h(x_1,\dots,x_n)=f(x_1,\dots,x_n)g(x_1,\dots,x_n),$$ where product in $RHS$ is product of loops. Is this the correct interpretation? What else could it be?
Thank you.
The intended multiplication induces the Pontryagin product on homology $H_{\bullet}(\Omega K)$, so it takes something in degree $n$ and something in degree $m$ and produces something in degree $n+m$. Given a singular $n$-cube $f : [0, 1]^n \to \Omega K$ and a singular $m$-cube $g : [0, 1]^m \to \Omega K$, their product is
$$f \times g : [0, 1]^{n+m} \to \Omega K$$
given by the "external" product $f(x_1, \dots x_n) g(y_1, \dots y_m)$.