I'm currently reading Mac Lane's Categories for the working mathematician and I'm having some trouble with the two following exercises from part III.
- Find (from any given object) an universal arrow to the forgetful functor $\mathbf{Rng}\rightarrow\mathbf{Ab}$ that forgets multiplication (it is important to stress that $\mathbf{Rng}$ means rings with units)
- Prove the second isomorphism theorem for groups, that is $SN/N\simeq S/S\cap N$ for $S,N\subset G$, $N$ normal in $G$, using only universality.
For the first one, it is just that I am not aware of the name of the mathematical construction: I can guess that the ring $R_{G}$ constructed from $G$ is a kind of ring with a copy of $\mathbb{Z}$ plus all products of elements of $G$, with equivalence relations $(a+b)c=ac+bc$ and $(na)b=a(nb)$ for integer $n$.
(I know for example that if we forget addition instead of multiplication then we get the ring algebra $R[G]$)
For the second one I don't know how to characterise $SN$ and $S\cap N$ in an element-free fashion. I did manage to prove the third isomorphism theorem.
The name of your construction is the tensor algebra, and it's seen more often as the free $k$-algebra on a vector space than as the free ring on an abelian group, as we have here. Anyway, you're right about the construction: $R_G$ has underlying abelian group the direct sum of all the tensor powers $G\otimes...\otimes G$, with product given by concatenation.
The most obvious characterization of $SN$ and $S\cap N$ is as the coproduct and product in the poset of subgroups of $G$. See if that works for you.