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.

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