Relation between R[X] and R[X,Y] as R-modules

Question

In my lecture notes it was left as an exercise to show that for any $R$ $R[X,Y]\simeq R[X]$ as R-modules. To show this you need a bijective function $\phi$ such that $\phi (x+y)=\phi(x)+\phi(y)$ and $\phi(rx)=r\phi(x)$. I don't know how to do this or if its even right since it seems hard to construct phi in such a way not to loose information when going from some $f_0(x,y)$ to $f_1(x)$.

Answer 1

As $R$-modules, $R[X]$ has a simple basis $\{1, X, X^2, \ldots\}$ and $R[X,Y]$ has $\{X^n Y^m \mid n \ge 0, m \ge 0\}$ (where $X^0 = Y^0 =1$ of course etc.) as a basis.

Two $R$-modules with countable bases are isomorphic. Just fix your favourite bijection between the bases and extend $R$-linearly..