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 ${A mathord{left/ {vphantom {A {rleft( M right)}}} right. kern-0em} {rleft( M right)}}$ -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.