Tensor Product II

Intro to Tensor Products review.

Let $R$ be a comutative ring with 1 and $A$ an $R$-module with a bilinear operation $m:A×A\rightarrow A$ given by $m(a,b)=ab$ for $a,b∈A$.
We know that $m$ is associated to a $R$-module homomorphism
$μ:A\otimes_R A \rightarrow A$.
Let the abuse of notation $(A\otimes_RA)\otimes_R A=A\otimes_R(A\otimes_RA)$ be true and let $Id_A:A\rightarrow A$ be the identity function in $A$.

  • Show that $m$ is an associative operation if and only if                       $\mu \cdot ( \mu \otimes Id_A)=\mu  \cdot (Id_A\otimes \mu)$


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