Prove that $K$ is not equal to $IJ$

Question

Consider the ring $\mathbb{Z}[x]$. Let $I = \{ f(x) : f(0)=0 \}$ and $J = \{ g(x) | \text{ all coefficients of $g$ are even} \}$.

Let $K \subset \mathbb{Z}[x]$ be $\{ ab: a \in I, b \in J \}$. Show that $K \neq IJ$.

I'm not sure how to prove this. Maybe i can show that $IJ$ is bigger than $K$, and show a counterexample where something is in $IJ$ but not in $K$? but not sure how to do this either.

Answer 1

Hint:

For example, $\;p(x)=4x^2+2x\in IJ\;$ , yet $\;p(x)\notin K\;$ . Try to prove this.