This is really two questions. First, consider a normal toric variety $X_\Sigma$. Its homogeneous coordinate ring $$R=\mathbb C[x_1,...,x_{|\Sigma(1)|}]$$ is graded by $A_{n-1}(X)$. In analogy with projective space, I guess that there is an analogue of the Proj construction: homogeneous ideals of $R$ not contained in the Stanley-Reisner ideal $B(\Sigma)$ of the fan $\Sigma$.

If $X_\Sigma$ is projective, is $X_\Sigma = Proj_{B(\Sigma)}(R)$?

Assuming this is true, I am curious about the case when $X_\Sigma$ is quasi-projective; $R$ has non-trivial elements in "negative" degree.

If $X_\Sigma$ is quasi-projective, is there a analogue of Proj with $X_\Sigma = QProj_{B(\Sigma)}(R)$?