Rings whose modules are projective over endomorphism rings

Let $R$ be either an Artin ring or a commutative ring. We show that every $R$-module is projective over its endomorphism ring if and only if $R$ is uniserial. We study rings, all of whose modules are projective over their endomorphism rings. We show that for Artin or commutative rings this is equivalent to being uniserial. We remark that we know of no examples other than uniserial rings and conjecture that all such rings are uniserial. Our proof uses the complete ring of quotients as in [4]. Our work is partially motivated by the work of Sally and Vasconcelos on commutative rings whose ideals are projective over their endomorphism rings [7]. We use $J$ throughout to denote the Jacobson radical and $\operatorname {Soc} (R)$ to denote the socle of $R$.