Fundamental theorem of Symmetric polynomials:
Let $R$ be a commutative ring and $e_0,...,e_n$ be the elementary symmetric polynomials of $R[X_1,...,X_n]$.
Let $\Phi:R[X_1,...,X_n]\rightarrow R[X_1,...,X_n]$ be the morphism of rings and $R$-modules such that $\Phi(X_i)=e_i$
Then $\Phi:R[X_1,...,X_n]\rightarrow R[X_1,...,X_n]^{S_n}$ is bijective.
Does this also hold in an arbitrary ring $R$?
Wikipedia states the theorem when $R$ is commutative, but I couldn't find which part of the proof given there uses commutativity of $R$ when I checked the proof carefully.