Following Lam's notation, a ring (with identity) $R$ is called Dedekind-finite if $ab=1\iff ba=1$ in $R$.

There are a lot of result about left invertible implies right invertible. But the results all require some finiteness property on the ring or the matrix ring. I am asking a proof or a counterexample of that that $R$ is Dedekind-finite impies that the matrix ring $\mathbb{M}_n(R)$ is Dedekind-finite.