Let $k$ be a field and $A$ and $B$ be two commutative $k-$algebras.
Furthermore, let $I$ be an ideal in $A$ and $N$ be a $A\otimes_kB$-module.
Then is it true that $((A/I) \otimes_k B) \otimes_{A\otimes_k B} \ N)$ as a $B-$module is isomorphic to $(A/I) \otimes_A N$?
Here the $B-$module structures shall both times be induced by the $A\otimes_kB$-module-structure of $N$.