AUTHOR: Apostolos Beligiannis ADDRESS: Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece. ABSTRACT: Our aim in this paper is to investigate and compare various algebraic homotopy theories defined in a natural way in a module category. We apply our results to Gorenstein rings. CURRENT STATUS: Preprint. To appear in Mathematica Scandinavica.