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Let $X_{\Sigma }$ be a normal complex toric variety whose toric rays span $N_{\mathbb{R}}$, so it has no torus factor. Form the Cox ring $S=\mathbb{C}[x_{\rho }]$, its toric irrelevant ideal $B_{\Sigma }$, and the diagonalizable groupThe divisor class group grading defines the action of $G$. The Cox construction recovers $X_{\Sigma }$ as the categorical quotient of $\mathrm{Spec}S\setminus V(B_{\Sigma })$ by $G$. It is a geometric quotient exactly when the toric fan is simplicial.

For the projective line, $S=\mathbb{C}[x_0,x_1]$, $B=(x_0,x_1)$ and $G={\mathbb{C}}^{\ast }$ acts by simultaneous scaling, giving $({\mathbb{A}}^2\setminus \{0\})/{\mathbb{G}}_m={\mathbb{P}}^1$. The ray-span condition matters: a positive-dimensional algebraic torus has a zero toric fan and no toric ray variables, so its construction needs the omitted torus factor as well.^[ref]

For a normal complex toric variety with no torus factor, its Cox ring isThere is one variable per toric ray, and the grading is by the divisor class group, rather than necessarily by ordinary integers. The degree-$[D]$ part corresponds to sections of the divisorial sheaf ${\mathcal{O}}_X(D)$ after choosing a representative toric divisor.

For ${\mathbb{P}}^n$, this is the usual polynomial ring with every variable in degree one. For a well-formed weighted projective space, the class group is $\mathbb{Z}$ and the variable degrees are its weights. When an algebraic torus factor is present, its invertible Laurent coordinates must be included separately; the ray-variable ring alone cannot recover a positive-dimensional algebraic torus.^[ref]