Interpretation functors which are full on pure-injective modules with applications to 𝑅-torsion-free modules over 𝑅-orders

Let R , S R,S be rings, X ⊆ mod \mathcal {X}\subseteq \text {mod} - R R a covariantly finite subcategory, C \mathcal {C} the smallest definable subcategory of Mod \operatorname {Mod} - R R containing X \mathcal {X} and D \mathcal {D} a definable subcategory of Mod \operatorname {Mod} - S S . We show that if I : C → D I:\mathcal {C}\rightarrow \mathcal {D} is an interpretation functor such that I X ⊆ mod I\mathcal {X}\subseteq \text {mod} - S S and whose restriction to X \mathcal {X} is full then I I