Polynomial ring and the free algebra

Question

In the Algebra book of Mac Lane there is an exercise in Chap. IV which tells me to construct a polynomial ring $A[X]$ for any set (not necessarily finite) $X$ ($A$ a ring), and to give correct the universal property. As far as the construction is concerned, I have no clue, but I suggest the following UMP: $A[X]$ is the free object in $\mathbf{A-alg}$ on $X$ (a fancier way to say this is to say that the "polynomial" functor is left adjoint to the forgetful functor $\mathbf{A-alg}\rightarrow\mathbf{Set}$, innit?). Do you agree with me?

Answer 1

This is indeed so; in fact Bourbaki (Algebra II, chap. 4, sect. 1 'Polynomials and rational functions) defines the polynomial ring $A[(X_i)_i\in I]$ to be the free commutative algebra on $I$.

Answer 2

If I give a set map from X to an A-algebra R, then it extends uniquely to a map of A-algebras from A [X] to R, so you have the universal mapping property right (the description as adjoint to the forgetful functor -- I am not sure what "free" means in this context, but you are probably right about that too).