Let $I_1, I_2,$ and $I_3$ be the principle ideals of the $R[x]$ generated by the polynomials $2, x+1, x^2+x+1$ respectively. Which of the ideals are maximal ideals of $R[x]$ if:
a) $R[x] = \mathbb{Z}$
b) $R[x] = \mathbb{Z_3}$
For b, I got that its only (x+1) because 2 is a unit and the other polynomial is reducible.
I'm pretty confused on a though since its not a field. I think the way to show this is just from the definition of maximal ideal aka showing that the ideal is properly contained in another ideal, but I'm lost on how to do that and which ones ewould even be maximal. Here is my attempt: $I_1 = (2)$, $\mathbb{Z}$\ $2$ is not a field so $I_1$ is not maximal But I'm really stuck on the two polynomials $x+1$ and $x^2+x+1$. If anyone could please post a full answer I'd really appreciate it. I need to know this for class tonight, but can't post a bounty until after it starts