Maranda’s theorem for pure-injective modules and duality

Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by π Let Λ be an R-order such that QΛ is a separable Q-algebra.Maranda showed that there exists k ∈ N such that for all Λ-lattices L and M, if L/L πk ≈ then L ≈ M Moreover, if R is complete and L is an indecomposable Λ-lattice, then L/L πk is also indecomposable. We extend Marandafs theorem to the class of R-reduced R-torsion-free pure-injective Λ-modules. As an application of this extension,we showthat if Λis an order over a Dedekind domain R with field of fractions Q such that QΛ is separable then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of Λ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of Λ. Further, with k as in Maranda'fs theorem, we show that if M is R-torsion-free and H(M) is the pureinjective hull of M then H(M)/H(M) πk is the pure-injective hull of M/Mπk. We use this result to give a characterisation of R-torsion-free pure-injective Λ-modules and describe the pure-injective hulls of certain R-torsion-free Λ-modules