I have some difficulties to answer at that question: "Describe who is the free commutative $k$-algebra on a set $X$, where $k$ is a commutative ring"
Actually I know the answer: the problem is to give a formal demonstration of the fact that the free commutative $k$-algebra on $X$ is $K[X]$, the set of all polynomials with coefficients in $k$ and commutative indeterminates in $X$.
I find this demonstration on internet, but I don't understand the last two lines, when he says : "Since $A$ is commutative, $\hat{f}$ factors through $p$, in the sense that there exists a map $f :K[X]→A$ such that $pf = \hat{f}$." The link where I found the demonstration is that one:
https://planetmath.org/FreeCommutativeAlgebra
Is there someone who can explain me this passage of the demonstration, or can give me another proof of that?