# Find prime ideals of Quotient Ring

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.

• 4. $x^2+1=0$ in $R$, so $x$ is invertible and hence $(x)=R$. – user26857 Apr 15 '19 at 21:29

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 $$(x+i), (x-i)$$.