Question: Let Ring be the category of rings and CRing be the category of commutative rings. Find a functor $F: \textbf { Ring } \rightarrow \textbf { CRing }$ such that if $R$ is commutative then $F(R)\cong R$.
I have two idea.
One is to construct a forgetful functor $G$ from Cring to Ring. It forgets the commutativity of any object in Cring. Therefore, $F$ is just the left adjoint functor to this forgetful functor. However, how do I find this left adjoint functor? It is not easy to find such bijective map between Cring$(F(R),R')$ and Ring$(R,G(R'))$.
Another one is to make $F$ to be the abelianisation of Ring. However, I have no idea on making a ring into a commutative ring, what I know is just for group.
Do I get wrong ideas?