So, the next, and hopefully last, question in my growing list of questions about adjoints to forgetful functors concerns the left adjoint to functor $U:\mathbf{Ring}\to\mathbf{AbGrp}$.
My approach so far has been to take the free monoid $F(A)$ over the set under the abelian group A, then the free abgroup $AbF(A)$ of $F(A)$, and finally identify all elements $'ab+ab'$ with $a(b+b)$ and $(a+a)b$.
But it does feel like I've gone too far in forgetting the abelian structure when making the free monoid. Perhaps we can take a free monoid over the group immediately and define it to be consistent with addition so that $a(b+c)d= abd+acd$.
I found that a similar approach was necessary when defining the free group over a monoid, but the situation is very different here, and I lack the skill to confidently pursue my own solutions.
Thanks in advance for any answers.