Let $R$ be a commutative ring. Using Quillen's $+$-construction, it is relatively easy to see that the algebraic K-theory of $R$, $K_*(R)$, admits a graded commutative product $$K_i(R)\otimes K_j(R) \to K_{i+j}(R).$$ The proof essentially boils down to showing that $BGL(R)^+$ is an H-group.
We also know that the $+$-construction agrees with Waldhausen's $S$-construction in this case.
My question is: Is there a way to construct this product on $K$-theory using Waldhausen's $S$-construction?
I know that this is done for commutative ring spectra in EKMM, so I guess I could apply their method to $HR$. But there is probably a much simpler way to see this, without having to pass by stable homotopy theory.
See Waldhausen's paper in the Annals, Generalized Free Products, section 9, for a construction of products using the Q construction. The same thing works for the S construction, the idea being that the tensor product of two filtered modules is a bifiltered module; I don't know a good reference for that at the moment.