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Are there group objects in:
If so, why doesn't anyone talk about them?
On the other hand,
$$ \begin{align} cogroup \ objects \ in \ \text{CRing} &= co(group \ objects \ in \ \text{AffScheme})\\ &= co(algebraic \ groups)\\ &= Hopf \ algebras \end{align} $$
Dual to my original question: $$ cogroup \ objects \ in \ \text{Scheme}=? $$
Well, any group object in $\mathbf{Ring}$ will be a group object in $\mathbf{Ab}$, and it is straightforward (using the Eckmann–Hilton argument) to show that every object in $\mathbf{Ab}$ admits the structure of a group object in a unique way: the new group operation will coincide with the old group operation. Thus any object in $\mathbf{Ring}$ has at most one group object structure.
But the fact is that there are only trivial group objects in $\mathbf{Ring}$. Indeed, let $R$ be a ring. Then every homomorphism $1 \to R$ is an isomorphism, so the only pointed objects in $\mathbf{Ring}$ are the trivial ones. In particular, the only group objects in $\mathbf{Ring}$ are the trivial ones.
