Proving membership of monomials to an ideal

Question

I am a grad student having background in Algebra. I need help with the following.

Let $I=\langle x_1^3,x_2^3,x_3^3,x_1 x_2 x_3,x_1^2 x_2^2, x_1^2 x_3^2, x_2^2 x_3^2 \rangle$ be an ideal in a polynomial ring $\mathbb{R}[x_1,x_2,x_3]$.

$\textbf{Question:}$ Prove that all monomials of degree greater than equal to $4$ belongs to the ideal $I$, i.e. $x_{1}^{\alpha_1} x_{2}^{\alpha_2} x_{3}^{\alpha_3} \in I$, whenever $\alpha_1 +\alpha_2+\alpha_3\geq 4$ and $\alpha_1 ,\alpha_2,\alpha_3 \in \mathbb{N}\cup \{0\}$.

Any help would be appreciated. Thank you.

Answer 1

It follows directly from the definition of ideals. By definition, an ideal absorbs the product with each element of the ring. So, when you have a monomial $x_1^ax_2^bx_3^c$ with degree $\geq 4$ you can just "complete" one of the generators by multiplying it with the remaining part.

You can use induction to prove it:

• Base case: just try all possible cases by hands (up to multiplicative constants, the monomials of degree 4 on three variables are few).
• Inductive case: if by inductive hypothesis all monomials of degree $n$ belong to $I$, you can just write a monomial of degree $n+1$ as $x_i\cdot m$ for $i \in \{1,2,3\}$, where $m$ is a monomial of degree $n$, so it belongs to $I$ because of the closeness under product