Suppose $A$ is a ring, $M$ is a right $A$-module, and $N$ is a left $A$-module.
In this situation, we can form the tensor product, $M\otimes_A N$, and this is an abelian group (and even a $Z(A)$-module where $Z(A)$ is the center of $A$).
On the other hand, we can look at the opposite ring, and then look at $N \otimes_{A^{op}} M$ which is a well defined abelian group (Z(A)-module).
Am I correct that the map $m\otimes n \mapsto n\otimes m$ is an isomorphism between these $Z(A)$-modules?
Is there anything missing or is this really trivial?