First a little introduction and notation. I have a question with a words that the book says.
Let k be a field. Let F $\in $ $k[x_1,...,x_n]$ We define the locus $V(F)$ of F , by $ V\left( F \right) = \left\{ {P = \left( {a_1 ,...,a_n } \right):F\left( P \right) = 0} \right\} \subset k^n $ Now consider the ring $ A = k[x1,..,x_n]/(F) $ Then note that an element $ g\in$ A defines a k-valued function on $V(F) $ indeed, if g is the class in A of a polynomial g´ \in k\left[ {x_1 ,...x_n } \right] then for x $\in$ V(F) the value g(x) = g´(x) does not depend on the choice of g´.
My question comes in this paragraph.
If F has no multiple factors , say $ F$ = $\prod f_i$ with $f_i$ $\nmid $ $f_j$ then it can be shown that F generates the ideal of all functions vanishing on $ V(F)$ ( This result is not my question ) . It follows from this that an element g$\in$ A is uniquely determined by the corresponding function g: X $\to$ k .
My question is about this, I know that must be so simple but I don´t know D:.