Follow 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 couterexample of that that $R$ is dedekind-finite impies that $\mathbb{M}_n(R)$ is dedekind-finite.