Let $I$ be an ideal of a ring. I know that $I$ is maximal if and only if $R/I$ is a field.
There's an example I'm working with, $(x^2+1) \subset \mathbb{C}[x]$ which is clearly not maximal because $(x^2+1) \subset (x+i)$.
However, if we consider $\mathbb{C}[x]/(x^2+1) \cong \mathbb{C}[i] = \mathbb{C}$ and $\mathbb{C}$ is a field, which should mean that $(x^2+1)$ is indeed maximal. It's clear that $\mathbb{C}[x]/(x^2+1) \not \cong \mathbb{C}$, but then what is that ring?
I suppose my question is, what ring is $\mathbb{C}[x]/(x^2+1)$?
Apply the Chinese remainder theorem to get $\mathbb{C}[x] / (x^2+1) = \mathbb{C}[x] / (x+i) \times \mathbb{C}[x] / (x-i) = \mathbb{C}^2$.