This question was inspired by this one. Given two ideals $A,B$ in a finitely generated commutative ring $R$. Is it possible to decide whether $A\cap B=AB$? Here $R$ is given by generators and relations, i.e. as a factor-ring of the free commutative ring over a finitely generated ideal, $A$, $B$ are given by their (finite) generating sets. I do not remember seeing this question in standard books on Groebner bases, but perhaps it is hidden there.
I think this problem can in fact be handled by Gröbner basis theory in the case $A$ is a polynomial ring. Since $I\cdot J \subseteq I\cap J$ for any two ideals, one can simply compute a Gröbner basis of $I\cap J$ (which is computed as the elimination ideal $( t\cdot I+(1-t)\cdot I ) \cap A$) and then checking whether each generator belongs to $I\cdot J$ (again using Gröbner basis algorithm).
EDIT: As Mark points out in his comment, this argument can be used to solve the problem for the general case $A=k[X]/R$ by considering the ideals $I+R$ and $J+R$ in $k[X]$.