The free commutative ring on a set $X$ is the polynomial ring with variables the elements of $X$. This polynomial ring is the free (additive) abelian group on the free (multiplicative) abelian monoid on $X$.
On the other hand, I've often seen a commutative ring defined as an abelian monoid object in the category of abelian groups, so I might have expected the composition to be the other way around - the free (multiplicative) abelian monoid on the free (additive) abelian group on $X$.
Does this give the same result? Is a ring equivalentely an abelian group object in the category of abelian monoids?