dimension of a coordinate ring Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?

Here is a positive result (see Fulton, Algebraic curves, Corollary I.7.4 on page 23). Let $k$ be some algebraically closed field. If $I \subseteq k[x_1,\dotsc,x_n]$ is an ideal, then $V(I) \subseteq \mathbb{A}^n(k)$ is finite if and only if $k[x_1,\dotsc,x_n]/I$ is a finite-dimensional $k$-vector space. In this case, we have $|V(I)| \leq \dim_k(k[x_1,\dotsc,x_n]/I)$.
A more precise result is that that $k[x_1,\dotsc,x_n]/\sqrt{I} \cong \prod_{P \in V(I)} k$, in particular $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$.
