This is one of the exercises in Maclane's book.

I can describe the following construction, which I think fits the bill.

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.