For a local ring $R$, we have a bijection $\mathbf P_R^n(R) \cong (R-\left\lbrace 0\right\rbrace)/R^\times$. This allows us to refer to points $\text{Spec} R \to \mathbf P_R^n$ using homogeneous coordinates. I know the same is not true for general rings $R$.
However, I am wondering about the case where we have an inclusion of local rings $R\hookrightarrow S$. Do we have a bijection $\mathbf P_R^n(S) = (S-\left\lbrace 0\right\rbrace)/S^\times$? According to the comments we have $\mathbf P_R^n(S) \cong \mathbf P_S^n(S)$ but it is unclear to me why this is true.
I have seen homogeneous coordinates being used on $\mathbf P^n(V)$ and $\mathbf P^n(K)$ for arbitrary valuation rings $V$ and $K=\text{Frac} V$, but I am unsure how this is justified.