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

Let R,S be rings, 𝒳⊆ mod−R a covariantly finite subcategory, 𝒞 the smallest definable subcategory of Mod-R containing 𝒳 and 𝒟 a definable subcategory of Mod-S. We show that if I:𝒞⟶𝒟 is an interpretation functor such that I𝒳 ⊆ mod-S and whose restriction to 𝒳 is full then $I$ is full on pure-injective modules. We apply this theorem to an extension of a functor introduced by Ringel and Roggenkamp which, in particular, allows us to describe the torsion-free part of the Ziegler spectra of tame Bäckström orders. We also introduce the notion of a pseudogeneric module over an order which is intended to play the same role for lattices over orders as generic modules do for finite-dimensional modules over finite-dimensional algebras.