Toric ideal

Remarks

The ideal $I_A$ is prime, since $k[x_1, \ldots, x_n]/I_A$ is isomorphic to a subring of a domain. Let $A$ also denote the $d \times n$ integer matrix with columns $a_i$. Then $I_A$ is generated, as a $k$-vector space and hence as an ideal, by the binomials $x^u - x^v$ with $u, v \in \N^n$ and $Au = Av$.

Example: $d = 2$, $n = 3$ and $a_1 = (1, 0)$, $a_2 = (1, 1)$, $a_3 = (1, 2)$, so $\varphi(x_1) = t_1$, $\varphi(x_2) = t_1 t_2$, $\varphi(x_3) = t_1 t_2^2$. Since $\varphi(x_1 x_3) = t_1^2 t_2^2 = \varphi(x_2^2)$, the binomial $x_1 x_3 - x_2^2$ lies in $I_A$, and in fact $I_A = (x_1 x_3 - x_2^2)$.

Non-example: if the characteristic of $k$ is not $2$, the binomial ideal $(x^2 - y^2)$ of $k[x, y]$ is not a toric ideal. It is not prime, because $(x - y)(x + y) \in (x^2 - y^2)$ while neither factor is.