Consider the functor $Rings\rightarrow Groups$ which sends a ring $R$ to its group of units $R^\times$. I am trying to show that this functor is co-representable, i.e. naturally isomorphic to $Hom(R,.)$ for some ring $R$. To do this, I tried to set up a correspondence between $Hom(R, A)$ and $A^\times$ for an arbitrary $A$ and some ring $R$, but I cannot seem to find a ring $R$ for which such a correspondence exists. Does anyone have any suggestions?
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