Find prime ideals of Quotient Ring

Question

Let $R$ be the ring $\Bbb C[x]/(x^2+1)$. Pick the correct statements from below:

1. $\dim_\Bbb C R=3$.
2. $R$ has exactly two prime ideals.
3. $R$ is a UFD.
4. $(x)$ 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.

Answer 1

As you said $$R=\mathbb C[x]/(x^2+1)\cong\mathbb C[x]/(x-i)\times\mathbb C[x]/(x+i)\cong \mathbb C\times\mathbb C$$

Now,

1. $R/P$ is an integral domain $\iff P$ is a prime ideal
2. $R/P$ is a field $\iff P$ is a maximal ideal

For $R$ to be field or integral domain you need to define a map $$R/P\rightarrow \mathbb C$$ This is accomplished by either one of the projections:

1. $(z,w)\mapsto z$
2. $(z,w)\mapsto w$

Which corresponds to the prime (and maximal) ideals $(x+i), (x-i)$.