I am studying 'The Homogeneous Coordinate Ring of a Toric Variety' by David Cox, and in the proof of his theorem 2.1 he defines $U_\sigma := \{x\in \mathbb{C}^{\Sigma(1)} : x^{\hat{\sigma}}\neq0\}$ where ${\Sigma(1)}\in \mathbb{N}$ and $x^{\hat{\sigma}}$ is a monomial in the ring of polynomials $S:=\mathbb{C}[x_i|i=1,..,{\Sigma(1)}]$
The proof then states 'the coordinate ring of $U_\sigma$ is equal to the localisation of $S$ at $x^{\hat{\sigma}}$. Why is this the case? I have tried looking at the equivalence relations that define the coordinate ring and this localisation but I cannot see the connection.