TITLE: Categories for the working mathematician exercises III 1
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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.