I am reading Andreas Gathmann's Algebraic Geometry notes where following statement is given regarding the product of affine varities:
Let $X\subset \mathbb{A}_{k}^{n}$ and $Y\subset \mathbb{A}_{k}^{m}$ be two affine varieties. And I am supposed to prove $I(X \times Y)= I(X)+ I(Y)$ where $I$ is the ideal function.
My first issue is: shouldn't it be the extended ideal of $I(X)+I(Y)$ in $k[x_{1},\ldots ,x_{n},y_{1},\ldots , y_{m}]$ instead of $I(X)+I(Y)$.
Secondly, in proving this, $I(X)+I(Y)\subset I(X\times Y)$ is trivial. For the other side I can see for $f \in k[x_{1},\ldots ,x_{n},y_{1},\ldots , y_{m}]; \;\;f(p,y_{1},\ldots , y_{m})\in I(Y)$ and $f(x_{1},\ldots ,x_{n},q)\in I(X)\; \; \forall (p,q)\in X\times Y$. But I can't figure out how to complete this.