## A criterion for inclusion in LT(I)

Fix a monomial order on $R = F[x_1, \ldots, x_n]$ and suppose $G = \{g_1, \ldots,g_m\}$ is a Gröbner basis for the ideal $I$ in $R$. Prove that $h \in LT(I)$ if and only if $h$ is a sum of monomials each divisible by some $LT(g_i)$.

Since $G$ is a Gröbner basis, we have $LT(I) = (LT(g_1), \ldots, LT(g_m))$. Thus $LT(I)$ is a monomial ideal with monomial generators $LT(g_i)$. By this previous exercise, $h \in LT(I)$ if and only if each term in $h$ is divisible by some $LT(g_i)$.