TITLE: Ideals of the tensor product $R\otimes_{k} S$?
QUESTION [8 upvotes]: Let $R$ and $S$ be commutative rings over a field $k$. Let $I$ be an ideal of the tensor ring $R\otimes_{k} S$. It is true that there exist ideals $I_{1}$ and $I_{2}$ of $R$ and $S$ respectively such that 
$$
I=I_{1}\otimes_{k} I_{2}? 
$$
If this is not true, are there any description of $I$? What if we don't assume commutativity of one of rings?

REPLY [3 votes]: The simplest example of the setup is probably

$R = k[x]$
$S = k[y]$
$R \otimes_k S = k[x,y]$

and the simplest ideals of $R \otimes_k S$ are principal ideals. The first place to look for such a thing that is a counter-example would be to pick a generator that isn't obviously a product of something from $k[x]$ and something from $k[y]$.
(P.S. I think you can arrange for an $I_1$ and an $I_2$ such that the "inclusion" from $I_1 \otimes_k I_2$ to $R \otimes_k S$ isn't monic, so really it isn't an ideal)