How should i think a maximal ideal of $\mathbb{Z}_{n}[x]$ if $n$ is composite number?
For example, is it possible to say that $\mathbb{Z}_{6}[x]/(2,x)\cong\mathbb{Z}_{6}/(2)\cong\mathbb{Z}_{2}$?
I wonder how to generalize this property.
Give some advice. Thank you!
Since dividing out by a maximal ideal $I$ gives a field, $\Bbb Z_n/(I\cap \Bbb Z_n)$ must be $\Bbb Z_p$ for some prime $p$ that divides $n$. So $I\cap \Bbb Z_n$ must be $(p)$. And $\Bbb Z_p$ is a principle ideal domain, so over $\Bbb Z_p$, $I$ reduces to $(f)$ for some $f\in\Bbb Z_p$. And if $I$ is maximal, that means $f$ is irreducible.
So any maximal ideal in $\Bbb Z_n[x]$ is of the form $(f,p)$ where $p$ is a prime that divides $n$, and $f$ becomes irreducible in $\Bbb Z_n[x]/(p)\cong \Bbb Z_p[x]$.