## Every monomial generating set is a Gröbner basis

Let $I \subseteq F[x_1,\ldots,x_n]$ be a monomial ideal with monomial generating set $G = \{g_1,\ldots,g_m\}$. Use Buchberger’s Criterion to show that $G$ is a Gröbner basis for $I$.

Let $i < j$. Recall that $S(g_i,g_j) = \frac{M}{LT(g_i)} g_i - \frac{M}{LT(g_j)} g_j$, where $M$ is a least common multiple of $g_i$ and $g_j$. Since the $g_i$ are monomials, we have $LT(g_i) = g_i$. So in fact $S(g_i,g_j) = \frac{M}{g_i}g_i - \frac{M}{g_j} g_j$ $= M - M = 0$.

Certainly $I \neq 0$; since $S(g_i,g_j) = 0$ for all $i < j$, by Buchberger’s criterion we have that $G$ is a Gröbner basis for $I$.