I have a homework question that goes as follows:
Let $A: Rng \to Gp$ denote the "additive group" functor, i.e. Taking a ring $R=(R,+,\cdot, 0,1)$ to its underlying additive group and taking a ring homomorphism $f: R\to S$ to the same map $f$, views as a group homomorphism $(R,+,0)\to (S,+,0)$. Does $A$ preserve limits? Does $A$ preserve colimits? Give a proof or counter example.
I think it preserves limits. I use the reasoning that $A$ preserves products and equalizers, so therefore it preserves limits. My problem is that it seems intuitively clear to me why this functor would preserve products and equalizers, it almost seems "trivial" with this functor, but is it really? Can someone help me see where the non-triviality comes in?
And then my question with colimits is, do we just show that A preserves pushouts and coequalizers? My intuition would then lead me to believe that it would preserve these things, but if it does not can you give me some reasoning to help me see why it wouldn't preserve these things?