Let $R$ be a graded ring and $0 \neq f \in R_1.$ Let $S = R[f^{-1}]_0.$ I am tasked with showing that $R[f^{-1}] = S[x, x^{-1}],$ the ring of Laurent polynomials. I am just confused on whether $S = R[f^{-1}]_0$ refers to the group of degree 0 homogeneous elements in the graded ring $R[f^{-1}].$
Also, if $R = \oplus_{i \in Z} R_i$ then I proved that $R[f^{-1}] = \oplus_{i \in Z} R_i[f^{-1}].$ So I don't know how $S[x, x^{-1}]$ can have enough "information" about $R$ to be isomorphic to $S[x, x^{-1}]$ because $S$ is only $R_0[f^{-1}].$ Any hints would be appreciated.