# Toric ideals are generated by binomials. $V(x)$ gives affine $n-1$ space in affine $n$-space. $x$ is not a binomial, yet affine $n-1$ space is toric? [closed]

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?

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