# Isomorphism of tensor product

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 $N$.

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You tagged as commutative algebra but don't specify in the question whether you want to restrict to commutative algebras. – Rasmus Nov 7 '11 at 13:54
I do, see the edit. – Cyril Nov 7 '11 at 14:00
There are obvious candidates for the maps that will be an isomorphism is there is one. Have you tried checking they are well-defined and mutally inverse? – Mariano Suárez-Alvarez Nov 7 '11 at 14:54

$\left(\left(A/I\right)\otimes_k B\right)\otimes_{A\otimes_k B} N \cong \left(\left(\left(A/I\right)\otimes_A A\right)\otimes_k B\right)\otimes_{A\otimes_k B} N$ (since $A/I\cong \left(A/I\right)\otimes_A A$)
$\cong \left(A/I\right)\otimes_A \left(\left(A\otimes_k B\right)\otimes_{A\otimes_k B} N\right)$ (tactical use of associativity of $\otimes$)
$\cong \left(A/I\right)\otimes_A N$.