Prime Ideals of $\mathbb{C}^{n}$

Question

What are the prime ideals of the ring $\mathbb{C}^{n}$? I was thinking that it is $e_{i} : i \in [n]$, where $I = e_{i} = \mathbb{C}^{(i)}= (\mathbb{C} , \dots , 0 , \dots , \mathbb{C})$, where the $i^{th}$ copy of $\mathbb{C}$ is not in $e_{i}$ because taking quotients yields $\mathbb{C}$ which is a domain? Is this reasoning correct?

Answer 1

Let $e_i=(0,...,1,0...)$ $1$ at $i$-place $e_i.e_j=0, i\neq j$. There exists $i_0$ such that $e_{i_0}$ is not an element of $I$, otherwise since $I$ is a vector space, $I=\mathbb{C}^n$.

Let $i$ different of $i_0$, $e_ie_{i_0}=0$ implies that $e_i\in I$ since $I$ is prime.