How do I find a functor from the category $\mathbf{Ab}$ to $\mathbf{Rng}$?

Question

I don't think just mapping any abelian group to a ring with this same underlying abelian groups will do. There's the issue of the ring being not necessarily unique. What other way is there which works?

Answer 1

There are many ways to turn an abelian group into a ring, but the simplest and natural one is to take the zero multiplication. That is, having an abelian group $A$ define multiplication on it as $$\forall x,y \in A: \, x\cdot y = 0$$

This makes an rng from an abelian group. To make this into an actual functor, we should specify how it acts on morphisms. Let's just map all homomorphisms in $\operatorname{Ab}$ into the same functions, which appear to be rng homomorphisms, thanks to zero multiplication:

$$f(x\cdot y)=f(0)=0=f(x)\cdot f(y)$$