Let $R$ be the ring $\Bbb C[x]/(x^2+1)$. Pick the correct statements from below:
- $\dim_\Bbb C R=3$.
- $R$ has exactly two prime ideals.
- $R$ is a UFD.
- $(x)$ is a maximal ideal of $R$.
First option is wrong as the dimension of $R$ over $\Bbb C$ is $2$. For $3$rd option, $\Bbb C[x]/(x^2+1)$ is isomorphic to $\Bbb C×\Bbb C$. But $\Bbb C× \Bbb C$ is not a UFD. Hence 3 is wrong. Is my justification for $3$rd option right? I don't know how to find the ideals of quotient ring. Please help me in understanding second and fourth option.