Proposition

Let $I \subset k[x_1, \cdots, x_n]$ be an ideal with the property that for every $f = \sum c_{\alpha} x^{\alpha} \in I$, every monomial $x^{\alpha}$ appearing in $f$ is also in $I$. Show that $I$ is a monomial ideal.

Solution

Let $A = \{ \alpha \mid \sum c_{\alpha} x^{\alpha} \in I, c_{\alpha} \ne 0 \}$. Let $I_A$ be the ideal generated by $\{ x^{\alpha} \mid \alpha \in A \}$

• $I_A \subset I$ because $\forall \alpha \in A, x^{\alpha} \in I$.
• $I \subset I_A$ because, for every $f \in I$, each monomial of $f$ is in $I_A$.