It is known that if $R=End_k(V)$, with $V$ a finite dimension $k$-vector space then $R$ is projective as $R^e$-module, thus of projective dimension $pd_{R^e}(R)=0$. If $V$ is of infinite dimension does any one knows if it is possible for $pd_{R^e}(R)$ to be finite?

Actually I'm searching for Von Neumann regular rings with this property.