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I was reading of Shaferevich's book of Basic Algebraic Geometry 1. On page 25 Example 4 it says that if $X$ and $Y$ are any closed sets in $\mathbb{A}^n$ and $\mathbb{A}^m$ respectively then $k[X\times Y]=k[X]\otimes k[Y]$. Here $k[X]$ is the ring of regular functions on $X$ (Coordiante ring of $X$).

He defines $\phi :k[X]\otimes k[Y]\rightarrow k[X\times Y]$ by

$\phi (\sum_if_i\otimes g_i)(x,y)=\sum_if_i(x)g_i(y)$. This is well-defined and surjective $k$-algebra homomorphism. But I could not understand how it is injective.

If anyone explains me I will be very grateful. Thank you.

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    Hint : The coordinate rings are finitely generated. – 2017-01-31
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    @C.Dubussy In the book it says that it is enough to show if $\{f_i\}$ and $\{g_j\}$ are linearly independent in $k[X]$ and $k[Y]$ respectively then $\{f_i\otimes g_j\}$ are linearly independent in $k[X\times Y]$. I could not understand why it is enough to show. Here linearly independent means $k$-linear independent right ? Whereas the coordinate rings are finitely generated as $k$-algebra. – 2017-01-31

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