$FI$-modules over preadditive categories and torsion theories

We work with $FI$-modules over a small preadditive category $\mathcal{R}$, 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 $\mathcal{R}$.