I need to prove that the tensor product of $A-$modules is commutative, ie:
$M\otimes_AN\simeq N\otimes_AM.$
I tried to construct an isomorphism $\varphi $ of kind $ m\otimes n\mapsto n\otimes m$, but I think this is not an homomorphism, because:
$\varphi(m\otimes n+m'\otimes n')=\varphi((m+m')\otimes (n+n'))\\ \qquad\qquad\qquad\quad\quad =(n+n')\otimes (m+m')\\ \qquad\qquad\qquad\quad\quad=n\otimes m+n'\otimes m+n\otimes m'+n'\otimes m'\\ \qquad\qquad\qquad\quad\quad\neq n\otimes m+n'\otimes m'=\varphi(m\otimes n)+\varphi(m'\otimes n').$
I thank any ideia.
This equation is wrong: $ φ(m⊗n+m′⊗n′)=φ((m+m′)⊗(n+n′))$. The tensor product is bilinear.