Maximal ideal of $\mathbb{C}[x]/(x^2+1)$

Question

Let $R$ be the ring $\mathbb{C}[x]/(x^2+1)$ Is it true that:

1. $dim_{\mathbb{C}}R =3$
2. $(x) $ is maximal ideal of $R$

I am not familiar with the dimension in ring theory, is it really meaning vectorspace dimension? How to clarify this?

How $(x)$ is treated as a subset of $R$ here, to ask if it is maximal? I didn't got a clarity on these.

Answer 1

Well, the ring $R={\Bbb C}[x]/\langle x^2+1\rangle$ has the monomial standard basis $\{\bar 1, \bar x\}$, since $\bar x^2+\bar 1=\overline{x^2+1}=\bar 0$ in $R$, where $\bar a = a +\langle x^2+1\rangle$ is the residue class modulo the ideal.

By the isomorphism theorems for rings, the ideals in ${\Bbb C}[x]$ containing $\langle x^2+1\rangle$ correspond one-to-one with the ideals of $R$.

The maximal ideals of ${\Bbb C}[x]$ are of the form $\langle x-a\rangle$ with $a\in{\Bbb C}$. From here you get the 2nd answer.