I have been wondering this question while trying to comprehend adjoint functors and the various definitions. If you let
$$F:\mathbf {R\text - Alg}\to \mathbf {Ring}$$
be the functor that sends each R-algebra to itself viewed as a ring in $\mathbf {Ring}$ and each R-algebra homomorphism to the same map viewed as a ring homomorphism. Does this functor have a right adjoint? What about a left adjoint? (Does the forgetful functor from $\mathbf {Grp}$ to $\mathbf {Set}$ have a left adjoint?)
I have an intuition for what the functor should do if it did exist. It should take any ring and turn it into the most general R-algebra possible from this. I assume this most general will come as a universal property, but again I'm not sure what this would be.
I asked my professor at the end of the class the other day and he didn't have much time to think about it but suggested it may be
$$G(A):=R\otimes_{\mathbb Z}A\in Ob_{\mathbf {R\text - Alg}}$$
the functor that sends every ring $A$ to this tensor product viewed as an R-algebra. I don't really see how this is an R-algebra even. Can somebody please help me understand this abstract non-sense. Thanks.