I am reading the Fulton 'Algebraic Curves' and I am currently trying to understand the proof for the existence of the intersection number. It is defined to be
$I(P; F\cap G) = \text{dim}_k \left(\mathcal{O}_p(\mathbb{A}^2)/(F,G)\right)$.
One of the properties this number has to fulfill is
'I(P,F ∩ G) is a nonnegative integer for any F, G, and P such that F and G intersect properly at P. I(P,F ∩G) = ∞ if F and G do not intersect properly at P.'
The proof goes as follows:
'If F and G have no common components, I(P,F ∩G) is finite by Corollary 1 of §2.9.'
Corollary 1: $\dim_k (k[X1,...,Xn]/I) = \sum_{i = 1}^N \dim_k (O_{pi} /IO_{pi}$)
Now my problem is, how do I get from $\dim(\mathcal{O}_p(\mathbb{A}^2)/(F,G))$ to $\dim(\mathcal{O}_p(\mathbb{A}^2)/(F,G)\mathcal{O}_p(\mathbb{A}^2))$ ?