Prove that $R[X_1,\dots,X_n] \otimes R[Y_1,\dots,Y_m] \cong R[X_1,\dots,X_n,Y_1,\dots Y_m] $

Question

Prove that $R[X_1,\dots,X_n] \otimes R[Y_1,\dots,Y_m] \cong R[X_1,\dots,X_n,Y_1,\dots Y_m] $ (as R-algebras)

My attempt:

Defined $\phi: R[X_1,\dots,X_n] \times R[Y_1,\dots,Y_m] \to R[X_1,\dots,X_n,Y_1,\dots Y_m] $ by

$\phi(p(X_1,\dots,X_n),q(Y_1,\dots,Y_m))= p(X_1,\dots,X_n)q(Y_1,\dots,Y_m)$

$\phi $is well-defined and an R-bilinear map.

So it induces,$\Phi:R[X_1,\dots,X_n] \otimes R[Y_1,\dots,Y_m] \cong R[X_1,\dots,X_n,Y_1,\dots Y_m]$ as,

$\Phi(p(X_1,\dots,X_n)\otimes q(Y_1,\dots,Y_m))=p(X_1,\dots,X_n)q(Y_1,\dots,Y_m)$

Defined,$\psi:R[X_1,\dots,X_n,Y_1,\dots Y_m] \to R[X_1,\dots,X_n] \otimes R[Y_1,\dots,Y_m]$ by,

$\psi((r{X_1}^{l_1}\dots {X_n}^{l_n} {Y_1}^{k_1} \dots {Y_m}^{k_m})=r({X_1}^{l_1}\dots {X_n}^{l_n} \otimes {Y_1}^{k_1} \dots {Y_m}^{k_m}) $

Then I tried to see, $\psi \circ \Phi = id$ and $\Phi \circ \psi = id$.

Do these maps work or is there any dispute in the proof?

Thanks in advance for help!

Answer 1

The scheme of the proof is good, but plunging in those details is very complicated.

It's easier to prove that

if $A$ is a commutative $R$-algebra, then $A\otimes R[X]\cong A[X]$

and then do induction on $m$, with the well-known fact that $A[X][Y]=A[X,Y]$.