This is one of the exercises in Maclane's book.
I can describe the following construction, which I think fits the bill.Find, for any given abelian group , a universal arrow from to the forgetful functor which forgets the multiplicative structure.
My first guess is that should be a quotient of a suitably defined free object on the elements of . Take the free monoid on the elements of , and then take the monoid ring on the elements of . Then, let be the ideal of generated by elements of the form , where are the canonical images of via the obvious "inclusion" , with in . Then, if I am not mistaken, is the desired "universal ring" on the abelian group .
Does this seem correct? If we have an morphism of abelian groups , we can extend to a morphism of monoids by defining for any word in M (and by definition of the empty product, the identity of is mapped to the identity of ). Then we can extend this map to a morphism of rings by linearity. But there is no guarantee at this point that the additive structure imposed by will be respected, so we have to quotient out by the ideal described above. The resulting map seems to do the job.
In Maclane's book, the category is meant to represent the category of rings, with arrows the ring morphisms preserving identity elements. However, I can read on Wikipedia that this category is now called , and that denotes the category of pseudo-rings. This distinction seems reasonable. If this is the case, we should replace by above. I think that the construction of a "universal" pseudo-ring on would be the same as above, except that we would take the free semigroup on the elements of instead of the free monoid.
If you have any comments, or insight, or if you can suggest a simplification of the above construction, I would be glad to hear it.