TITLE: Tensor product of 2 coordinate rings
QUESTION [9 upvotes]: For the term variety, I mean the irreducible algebraic set.
My question is, if $V$ and $W$ are 2 varieties over a field $\Bbbk$, then does $\Bbb{k}[V]\otimes \Bbb{k}[W]$ has special structure?
I try to prove that, it is another coordinate ring of another variety, which depends on $V$ and $W$. Then from the irreducible of $V$ and $W$ I reduced the problem to understand the tensor product of $\mathfrak{R}/\mathfrak{p} \otimes \mathfrak{R}/\mathfrak{q}$, which is known that isomorphic to $\mathfrak{R}/\mathfrak{(p+q)}$, where $\mathfrak{R}$ is a Noetherian ring(or a finitly generated algebra), and $\mathfrak{p},\mathfrak{q}$ are prime ideals of $\mathfrak{R}$.
So, what can we imagine the element of $\mathfrak{R}/\mathfrak{p} \otimes \mathfrak{R}/\mathfrak{q}$? I just can understand the tensor product for 2 things(roughly speaking): making bilinear map into linear map, and playing the role of a funtor acting on exact sequence. I could not imagine concretely the element of tensor product(for example, $u\otimes v$ where $u,v$ are two vectors). 
Therefore, I could not define the tensor product of 2 coordinate rings( there may be some different ways, please answer here if you got them).
The above sentences describe my situation. Please help me point it out the intuitive picture of tensor product and help me to solve(prove/disprove) my initial question on tensor product of 2 coordinate rings.
Thanks.

REPLY [8 votes]: I wanted to say more about the approach suggested in Steve's answer. The first thing to show is that this is well defined, which is not too hard. It's surjective, since all of the coordinate functions on $V \times W$ are in the image.
Injectivity is less obvious, but remember that you have a field, over which all modules are free, on hand. If $\{f_i\}$ is a (gigantic) basis for $k[V]$ over $k$, and $\{g_j\}$ a basis for $k[W]$, then $\{f_i \otimes g_j\}$ is a $k$-basis for the tensor product. If you have some expression $\sum a_{ij}f_i\otimes g_j$, $a_{ij} \in k$, which maps to zero in $k[V \times W]$, then for any fixed $Q \in W$ the polynomial $\sum a_{ij}g_j(Q)f_i$ vanishes on $V$. What can you conclude?