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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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up vote 7 down vote accepted

Take $R=\mathbb Z[t,t^{-1}]{}{}{}{}$.

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The goal here is to find the free ring on a unit, so one starts with the free ring on an element and then forces that element to become a unit. –  Qiaochu Yuan Jun 18 '13 at 20:46

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