Monoid objects in abelian groups are rings

By [L1] and [L2], a monoid object structure on $A$ is a homomorphism $\mu :A{\otimes }_{\mathbb{Z}}A\to A$ and a homomorphism $\eta :\mathbb{Z}\to A$. The map $\mu$ corresponds to a bilinear operation $(x,y)\mapsto x\cdot y$ on $A$, and $\eta$ is determined by the element $1:=\eta (1)\in A$.

The monoid-object associativity and unit diagrams from [L1] translate under the correspondence in [L2] to $(x\cdot y)\cdot z=x\cdot (y\cdot z)$ and $1\cdot x=x=x\cdot 1$. Because $\mu$ is a group homomorphism in each variable, the operation is bilinear, hence distributes over the abelian-group addition on both sides.

Thus the additive abelian-group structure on $A$, together with multiplication $\cdot$ and unit $1$, satisfies exactly the axioms in [L3], so $A$ is a ring. Conversely, any ring yields such a bilinear multiplication and unit homomorphism, hence a monoid object in $\mathbf{A}\mathbf{b}$.

Therefore monoid objects in abelian groups are exactly rings.