Let $A, B$ be rings, let $M$ be a $A$-module, and $N$ a $B$-module. Let $\varphi: A\rightarrow B$ be a morphism of commutative rings, and use $\varphi$ to give $N$ the structure of an $A$-module.
I'm aware that there is a natural isomorphism
$$ M\;\;\cong\;\; M\otimes_{A} A $$ and that there is a certain ``associativity'' property that the tensor product satisfies:
$$ \big(X\otimes_{A} Y\big)\otimes_{A} Z\;\;\cong\;\; X\otimes_{A}\big(Y\otimes_{A} Z\big) $$
The question is: Does the associativity property hold when tensoring with respect to different rings? That is, is
$$ \big(M\otimes_{A} N\big)\otimes_{B} B\;\;\cong\;\; M\otimes_{A}\big(N\otimes_{B} B\big)$$ true?