## Every monomial generating set is a Gröbner basis

Let $F$ be a field. Suppose $I \subseteq F[g_1,\ldots,g_n]$ is a monomial ideal with monomial generating set $G = \{g_1,\ldots,g_m\}$. Prove that $G$ is a Gröbner basis for $I$.

Recall that $LT(I) = (LT(p) \ |\ p \in I)$. Let $p \in LT(I)$, with $p = \sum a_i LT(p_i)$. In particular, each $LT(p_i)$ is in $(g_j)$ for some $j$, so that $p \in I$. Hence $LT(I) \subseteq I$. Conversely, we have $g_i = LT(g_i) \in LT(I)$ for each $i$. Thus $I = LT(I)$. Since $LT(I) = (g_1, \ldots, g_m) = (LT(g_1), \ldots, LT(g_m))$, $G$ is a Gröbner basis of $I$.