The Structure of Modules over Hereditary Rings

Let A be a bounded hereditary Noetherian prime ring. For an A-module MA, we prove that M is a finitely generated projective \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${A \mathord{\left/ {\vphantom {A {r\left( M \right)}}} \right. \kern-\nulldelimiterspace} {r\left( M \right)}}$$ \end{document}-module if and only if M is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\pi }$$ \end{document}-projective finite-dimensional module, and either M is a reduced module or A is a simple Artinian ring. The structure of torsion or mixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\pi }$$ \end{document}-projective A-modules is completely described.