This question is a bit urgent, as I have written this in my Master's thesis and I have to submit it next week!
Proposition: Let $X_\Sigma$ be a toric variety without torus factors, meaning that the minimal generator of the rays $u_\rho$ generate the whole lattice $N$, then with $r=|\Sigma(1)|$ (the number of one-dimensional cones of the fan) we get the isomorphism $$ \operatorname{Cl}(X_\Sigma)\simeq \mathbb{Z}^{r-n} $$ Proof. We take the exact sequence from CLS ("Toric varieties" from Cox, Little, Schenck) Theorem 4.1.3 (p. 172) together with the remark directly above: $$ 0 \longrightarrow M\simeq \mathbb{Z}^n \longrightarrow \operatorname{Div}_{T_N}(X_\Sigma)\simeq \bigoplus_{\rho\in\Sigma(1)} \mathbb{Z} D_\rho\longrightarrow \operatorname{Cl}(X_\Sigma)\longrightarrow 0. $$ This simplifies to the short exact sequence $$ 0 \longrightarrow \mathbb{Z}^n \longrightarrow \mathbb{Z}^{r} \longrightarrow \operatorname{Cl}(X_\Sigma)\longrightarrow 0 $$ and with the homomorphism theorem, i.e. for any homomorphism $f:G\to H$ we have $G/\ker f \simeq \operatorname{im}f$ which yields the claim. $\square$
I don't think this is true, as CLS (and other resources) add a finite group to the divisor class group. If $\Sigma$ contains an $n$-dimensional cone, I have found out that $\operatorname{Pic}(X_\Sigma)\simeq \mathbb{Z}^{r-n}$, but only if $\Sigma$ is smooth, we have an equality with $\operatorname{Cl}(X_\Sigma)$.
Where is my logical error, or can someone provide a counterexample? This proof seems to be too simple that nobody else would already have found it.
