In his 1987 article *The geometry of toric varieties* Danilov gives a combinatorial presentation of Chow rings of complete smooth toric varietes. Given a complete unimodular fan $\Sigma$ we have
$$
A^*(X_\Sigma) \cong \mathbb Z[U]/(I+J),
$$
where $U$ is the set of rays in $\Sigma$, $I$ the Stanley-Reisner ideal of the associated simplicial complex (on the rays) and $J$ an ideal depending on the coordinates of the minimal ray generators.

In the recent preprint *Hodge theory for combinatorial geometries* by Adiprasito, Huh and Katz, the authors state that this isomorphism still holds when $\Sigma$ isn't complete. They cite the following two articles for this:

- Bifet, De Concini, Procesi,
*Cohomology of regular embeddings*, 1990 - Brion,
*Piecewise polynomial functions, convex polytopes and enumerative geometry*, 1996

However, I'm unable to extract this result from those articles. Maybe someone can point out why this is implied by the results in the cited articles or possibly knows another reference?