# Exercise of commutative algebra, rational functions.

This exercise is of my weekly newsletter of the subject of commutative algebra. My knowledge is restricted to the book of William Fulton, Algebraic Curves. I need help to solve it, any hints.

Exercise:

Let $Y \subseteq X$ two algebraic varieties and let $\mathcal{K}(X,Y)$ be a subset of $\mathcal{K}(X)$ that consists of rational functions defined at some point in $Y$. Prove that $\mathcal{K}(X,Y)$ is a local subring of $\mathcal{K}(X)$ and that its residual field is isomorphic to $\mathcal{K}(Y)$.

The ring in this case is $$\mathcal K(X,Y) = \{ f \in \mathcal K(X) \mid f \text{ is defined on } Y \}.$$
HINT What about those functions that are zero on $Y$?