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 $m:A×A?A$ given by $m(a,b)=ab$ for $a,b?A$.
We know that $m$ is associated to a $R$-module homomorphism
$?:A?_RA?A$.
Let the abuse of notation $(A?_RA)?_RA=A?_R(A?_RA)$ be true and let $Id_A:A?A$ be the identity function in $A$.

  • Show that $m$ is an associative operation if and only if                       $? ? (??Id_A)=? ? (Id_A??)$

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