Here $M$ and $N$ are both left $R$-module. I have seen that $M\otimes_R N \cong N \otimes_R M$ is meaningful only when $R$ is commutative, but I can't see the reason.
In the noncommutative case, tensor product of two left $R$-module $M,N$ could be defined as an left $R$-module$M\otimes_R N$, right(although it seems that it's useless)? And then we could ask if there always holds $M\otimes_R N \cong N \otimes_R M$ as left $R$-module. I think it's true but I can't see why this is meaningless. Could you give some hints? Thanks in advance.