An $R$-module $M$ is called quasi-projective if $\text{Hom}_R(M,M)\to\text{Hom}_R(M,N)$ is surjective for every surjective homomorphism $M\twoheadrightarrow N$.

What are the rings $R$ for which every quasi-projective $R$-module is projective? Does there exist such a ring which is not semisimple?