We say a homomorphism $f:A\rightarrow B$ of noetherian rings is non-degenerate if the induced map $f^*:{\rm Spec}(B) \rightarrow {\rm Spec}(A)$ maps ${\rm Ass}(B)$ into ${\rm Ass}(A)$.
Let $f:A \rightarrow B$ and $g:A \rightarrow C$ be homomorphisms of noetherian rings. Suppose
$B \otimes_{A}C$ is noetherian,
$f$ is flat,
$g$ is non-degenerate.
Then $1_{B} \otimes g:B \rightarrow B \otimes_A C$ is also non-degenerate.
I was trying to say that $f^*({\rm Ass}(B \otimes C)) \subset {\rm Ass}(B)$, but still stuck.