FI-modules over preadditive categories and torsion theories

Abstract We work with FI-modules over a small preadditive category R , viewed as a ring with several objects. Our aim is to study torsion theories for FI-modules. We are especially interested in torsion theories on finitely generated FI-modules and the category of what we call “shift finitely generated” FI-modules. We also apply these methods to study inductive descriptions of FI-modules over R .