The Structure of Modules over Hereditary Rings

Let $A$ be a bounded hereditary Noetherian prime ring. For an $A$-module $M_A$, we prove that $M$ is a finitely generated projective $A/r(M)$-module if and only if $M$ is a $\pi$-projective finite-dimensional module, and either $M$ is a reduced module or $A$ is a simple Artinian ring. The structure of torsion or mixed $\pi$-projective $A$-modules is completely described.