On systems of binomials in the ideal of a toric variety

Let $\Gamma$ be a toric set in the affine space ${\mathbb A}_k^n$. Given a set of binomials $g_1,\ldots,g_r$ in the toric ideal $P$ of $\Gamma$, we give a criterion for deciding the equality rad( $g_1,\ldots,g_r$) = $P$. This criterion extends to arbitrary dimension, and to arbitrary fields, an earlier result which concerned only monomial curves over an algebraically closed field of characteristic zero.