@R.vanDobbendeBruyn Thank you for the example. But it is of the same sort as the example $R=\mathbb{Q}[x]$ I gave in my post: my R and your L are both finitely generated over a subring and the homomorphism $f$ is then defined as $f(r)=rI$ on the subring and $f(g)$ a suitable non-diagonal matrix for the generator $g$. I'm more interested about a situation where $R$ isn't of this kind...