Proposition 1.1.11 of Cox-Little-Schenk's Toric Varieties states that an ideal $I \subseteq \mathbb{C}[x_1, \dots, x_n]$ is toric iff it is prime and generated by binomials. Setting $I = (x_1) \subset \mathbb{C}[x_1, \dots, x_n]$ gives that $I$ is prime, but not generated by binomials, yet $V(I) \cong \mathbb{A}^{n-1}\subset \mathbb{A}^n$ is toric. Is this a counterexample to Proposition 1.1.11 in Cox-Little-Schenk's Toric Varieties?

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    $\begingroup$ Your ideal $I$ is generated by the single binomial $x_1-0$. $\endgroup$ – Balazs Elek Jun 29 at 22:31
  • $\begingroup$ $x_1-0x_2$ is a binomial in two variables. $\endgroup$ – user73577 Jun 30 at 16:28