The group scheme formula_23 is isomorphic to the multiplicative group scheme formula_24 over any base, so for any commutative ring , the groups formula_25 and formula_26 are canonically isomorphic to formula_27. Note that the functor formula_24 (that is, formula_29) is representable in the sense: formula_30 for commutative rings "R" (this for instance follows from the aforementioned adjoint relation with the group ring construction). Explicitly this means that there is a natural bijection between the set of the ring homomorphisms formula_31 and the set of unit elements of "R" (in contrast, formula_32 represents the additive group formula_33, the forgetful functor from the category of commutative rings to the category of abelian groups).