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?



Module complements to rings embedded in a projective module
For (A) a ring $R$ is selfinjective if it is injective as an $R$-module (on e should say left or right). By definition of injective this means $R$ has a complement in any module. Examples include Frobenius and quasi-frobenius rings.


If $R$ is von Neumann regular, then any finitely generated submodule of a projective module has a complement. So again if $R$ embeds in a projective module there is a complement.


