Under a certain normalization assumption we prove that the $\Pro^1$-spectrum $\mathrm{ BGL}$ of Voevodsky which represents algebraic $K$-theory is unique over $\Spec(\mathbb{ Z})$. Following an idea of Voevodsky, we equip the $\Pro^1$-spectrum $\mathrm{ BGL}$ with the structure of a commutative $\Pro^1$-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over $\Spec(\mathbb{ Z})$. For an arbitrary Noetherian scheme $S$ of finite Krull dimension we pull this structure back to obtain a distinguished monoidal structure on $\mathrm{ BGL}$. This monoidal structure is relevant for our proof of the motivic Conner-Floyd theorem. It has also been used by Gepner and Snaith to obtain a motivic version of Snaith's theorem.
Categories: math.AG math.AT