Testing of maximal ideal

Question

This question was asked in NET. Let $R=\mathbb{C}[x]/(x^2+1).$ Is $(x)$ is a maximal ideal of $R?$ I think the question is invalid. Because the ideal of $R$ are of the form $A/(x^2+1).$ Where $A \subseteq (x^2+1)$ and $A$ is an ideal of $\mathbb{C}.$ Now $\mathbb{C}$ is a field so $\mathbb{C}[x]$ is a P.I.D. $(x)$ is an ideal of $R$ so, $(x) \subseteq (x^2+1).$ But $x^2+1$ is not a divisor of $x.$ Therefore, $(x) \nsubseteq (x^2+1).$Hence, $(x)$ is not even an ideal of $R$. Am I right? My 2nd question is what is $R/(x)=\frac{\mathbb{C}[x]/(x^2+1))} {(x)}?$ Please help me. Thanks in advance.

Answer 1

In the ideal generated by $x$ we find the element $-x \times x=-x^2\equiv 1 \bmod (x^2+1)$.

This means that the ideal generated by $x$ is the whole ring, because it contains $1$.

Note also:

Because $\mathbb C$ is algebraically closed every polynomial in $\mathbb C[x]$ splits and the only irreducible polynomials are the linear ones.

So that $x^2+1=(x+i)(x-i)$ is not irreducible in $\mathbb C[x]$. So you need to take some care - for example $(x+i)$ and $(x-i)$ are zero divisors in the factor ring.