If $R$ is a ring and $R^n$ and $R^m$ are isomorphic as left $R$-modules then they are also isomorphic as right R$-modules

Question

Is it necessarily true that if $R$ is a ring and $R^n$ and $R^m$ are isomorphic as left $R$-modules then they are also isomorphic as right $R$-modules.

It appears as if they are.

Answer 1

Yes. For this to have content, $R$ must be non-commutative, and non-Noetherian.

If $\phi:R^m\to R^n$ is a left $R$-module isomorphism, and $\psi:R^n\to R^m$ is its inverse, then they correspond to matrices $A$ and $B$ over $R$ with $AB=I_m$ and $BA=I_n$. But then $A$ and $B$ correspond to right $R$-module maps $R^n\to R^m$ and $R^m\to R^n$ which are inverse to each other.