When constructing the projective space scheme $\mathbb{P}_R^n$ for a ring $R$, we may take the subrings $$ A_i = R\left[\tfrac{X_0}{X_i}, \ldots, \widehat{\tfrac{X_i}{X_i}}, \ldots, \tfrac{X_n}{X_i}\right], \; i = 0, \ldots, n, $$ of the ring $A^{(0)}$of homogenous elements of $A = R[X_0, \ldots, X_n, X_0^{-1},\ldots, X_n^{-1}]$ and glue the affine schemes $U_i = \text{Spec}(A_i)$ together along the open subschemes $U_{ij} \subseteq U_i$ with $$ U_{ij} = D_{U_i}\bigl(\tfrac{X_j}{X_i}\bigr) \cong \text{Spec}\left(A_i\left[\left(\tfrac{X_j}{X_i}\right)^{-1}\right]\right) $$
I read that we can use the identity to glue, but why (and how) is $$ A_i\left[\left(\tfrac{X_j}{X_i}\right)^{-1}\right] \cong A_j\left[\left(\tfrac{X_i}{X_j}\right)^{-1}\right] $$ for $i \neq j$?