Proving that $M\otimes N \approx N\otimes M$ (help needed)

Question

This question is from my assignment in Tensor products and I need help in solving the problem. I have been following atiyah and macdonald.

Question : Prove that $M\otimes N\approx N\otimes M$.( Here M and N are modules)

I am not very good in Tensor products and here is what I thought:

Intuition is clear to me as if $C =A^{M\times N}$ and D is the submodule by Which I will factor C to get $M\otimes N$, then C is generated by elements which are additive and satisfy scalar multiplication wrt to both M and N. So, divison modulo it make $M\otimes N$ isomorphic to $N\otimes M$.

But for proving rigoriously , I need to find a map f which will map $M\otimes N \to N\otimes M$. This map will be from $A^{M\times N}/D \to A^{N\times M}/D$.

Can you please help me with the construction of such map?

I would try rest of proof by myself.

Thanks!

Answer 1

Hint: Use the universal property of the tensor product to construct linear maps $$M \otimes N \to N \otimes M: m \otimes n \mapsto n \otimes m$$ and $$N\otimes M \to M\otimes N: n \otimes m \mapsto m\otimes n.$$ Then show that these maps are inverse to each other.