Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module.
A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-module complement. But this is a very strong assumption. What are weaker but sufficient criteria for an $R$-module complement to exist.
B) What is a non-free example where $M$ does not admit an $R$-module complement in $M.
C) What is a non-free module example where $M$ does admit a complement?