A criterion for inclusion in a monomial ideal

Let I = (m_i \ | i \in I) be a monomial ideal in F[x_1,\ldots,x_n]. Prove that p \in I if and only if each term in p is divisible by some m_i. Use this criterion to show that p = x^2yz + 3xy^2 is in (xyz, y^2) but not in (xz^2, y^2).


Certainly, if each term in p is divisible by some m_i then p \in I. Suppose now that p \in I. We have p = \sum a_im_i, where a_i = \sum n_{i,j}. (each n_{i,j} is a monomial.) Then p = \sum_i \sum_j n_{i,j}m_i. Thus each term in p is divisible by some m_i.

Now let p = x^2yz + 3xy^2. Since x^2yz = x(xyz) and 3xy^2 = 3x(y^2), p \in (xyz, y^2). However, since xz^2 does not divide either x^2yz or 3xy^2, p \notin (xz^2, y^2).

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