On a subring of the Laurent polynomials ring

Question

Let $R$ be a commutative ring with identity and consider the Laurent polynomials ring $S:= R[x,x^{-1}]$ and a polynomial $f\in S$. Suppose that $f$ is invertible in $S$ and $f\in R[x+x^{-1}]$. Is it true that $f^{-1}$ belongs to $R[x+x^{-1}]$? The article I am reading claims that $R[x+x^{-1}]\cong R[y]$ but I can not see why. So please help me to understand this isomorphism.

Answer 1

$R[x+x^{-1}]$ is by definition the image of the morphism $$ \Phi:R[y]\to R[x,x^{-1}] $$ determined by $\Phi(y)=x+x^{-1}$. We have to show that this morphism is injective. Else, there would exist a $P\ne 0$ in $R[y]$ such that $P(x+x^{-1})=0$. But now let us consider the principal term of $P$ in $P=c_ny^n+$ lower terms, $c_n\ne 0$ in $R$, then $P(x+x^{-1})=c_nx^n+$ lower terms, this is not zero, so $\Phi(P)\ne 0$, contradiction.