Let $A$ and $B$ be algebras over a field $K$. The ideals of the tensor product $A\bigotimes_K B$ are of the form $I\bigotimes_K J$ where $I$ is an ideal of $A$ and $J$ is an ideal of $B$?

4$\begingroup$ No. The polynomial ring $K\left[X,Y\right]\cong K\left[X\right]\otimes K\left[Y\right]$ over a field $K$ should give you a good hint about how complicated the ideals of a tensor product can get. $\endgroup$ – darij grinberg Jan 20 '12 at 15:28

2$\begingroup$ For an even more complicated situation, consider $K[[x]]\otimes K[[y]]$, a tensor product of two noetherian rings which results in a nonnoetherian ring. $\endgroup$ – the L Jan 20 '12 at 15:30

4$\begingroup$ Let $Kk$ be a finite extension of fields. Comparing $k$dimensions shows that multiplication $K \otimes_k K \to K$ must have nontrivial kernel which thus cannot be of the form you expected. $\endgroup$ – Ralph Jan 20 '12 at 15:38

2$\begingroup$ Note to people downvoting the question: it is not a completely wild guess. See, e. g., Lemma 2.7 in Chapter IV of Milne's Class Field Theory ( jmilne.org/math/CourseNotes/cft.html ) for a case when it is true. $\endgroup$ – darij grinberg Jan 20 '12 at 15:38

2$\begingroup$ Well I don't think that the questioner has considered any examples before posting this. See also the FAQ mathoverflow.net/faq. 1 $\endgroup$ – Martin Brandenburg Jan 20 '12 at 16:16