Let $R$ be a ring (with identity, but not necessarily commutative). If given the presumption that for any left free $R$-module $M$, every submodule of $M$ is free, what can we say about $R$?
Apparently any left principal ideal domain fulfills the above requirement and it is the only possibility for $R$ commutative. But I have no idea about the case where $R$ is noncommutative. Does anyone have some thought of this problem?