Algebraic $K$-theory as extraordinary homology theory on the category of associative rings with unity

Algebraic $K$-theory can be constructed by means of the homotopy groups of the abstract simplicial structure on the group of invertible matrices $GL(A)$ of the ring $A$. This structure may be naturally taken as two-sidedly invariant. Of basic interest is the multiplication in the functor so obtained, which for different rings $A$ assumes different aspects.