Group objects in the category of rings

Question

Are there group objects in:

• $\text{Ring}$
• $\text{CRing}$

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: $$ cogroup \ objects \ in \ \text{Scheme}=? $$

Answer 1

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.