Consider the ring $R[x,x^{-1}]$ where $R$ is a commutative ring. I want a description of the ring after completing at the ideal $(1+x)$.
So my guess is that it is isomorphic to $R[[y]]$, where the isomorphism is given by $y \to 1+x$. The reason I guessed this was because $x$ is already invertible after completing at the ideal $(1+x)$. So intuitively I think it should be same as $R[x]$ completed at the ideal $(1+x)$ and therefore the guess. Any help with proving this or any references are appreciated. Thanks in advance.
