Let's say that a (right) module $M$ is *well complemented* if every non-zero submodule of $M$ has an indecomposable direct summand (by the way, is there a better or more standard name for this property?). For instance, every module of finite uniform dimension is well complemented.

Question.Is the regular right module $R_R$ of a von Neumann regular ring $R$ well complemented?

As a recall, a ring $R$ is *von Neumann regular* if, for every $x \in R$, there exists $y \in R$ such that $x = xyx$.