This is an elementary question about ideals. Consider a ring homomorphism $ f: \mathbb{Z} \rightarrow \mathbb{Z}[x], $ and consider the ideal $\left< 2\right>$ in $\mathbb{Z}$. When why is it that $f(\left< 2\right>)$ is not an ideal?
Some websites say that $f(\left< 2\right>)$ is not an ideal because it does not contain nonconstant polynomials. That still doesn't make sense on why it is not an ideal.
Thank you all.