Exercise 21, § 7

Let $A$ and $A’$ be two noetherian rings, and let q (resp. q’) be an ideal of $A$ (resp. $A’$) contained in the radical of $A$ (resp. $A’$) and such that long(A/q) (resp. long(A’/q’)) is finite. Let $\Omega$ be the ideal $q \otimes A' + A \otimes q'$ of the ring $A \otimes_Z A'$. Prove that it is contained in the radical of $A \otimes_Z A'$ and that one has the equality