Hello I have the following problem:
Check if the forgetful functor from $R$-modules to abelian groups for a ring $R$ is exact or not.
As I understood our case we have the following functor $$F:Mod(R)\rightarrow Ab;~~~(M,0,1,\cdot,+)\mapsto (M,+,0)$$ Which takes as an input an $R$-Module and gives us the corresponding abelian group. But it does not only connect objects from the category, it also connects the morphisms in each category, i.e. we also have $$Hom(M,N)\rightarrow Hom(F(M),F(N));~~f\mapsto f$$ where $f:M\rightarrow N$ is a homomorphism between $R$-modules.
I don't know if everything is correct till now. But then I need to check if it is exact. So we had the following definition:
Let $R$ be a ring, a sequence of $R$-modules is a collection $(M^i)_{i\in I\subset \Bbb{Z}}$ of $R$-modules together with $R$-module homomorphisms $f^i:M^i\rightarrow M^{i+1}$ where $i,i+1\in I$. Then the sequence is exact if $Im(f^i)=Ker(f^{i+1})$.
But somehow I don't see what sequence I need to take in my case and what this has to do with a functor, I mean I only speak about $R$ modules in the definition of exact and not about functors. It would be very nice if someone could explain it to me a bit.
Thanks for your help.