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 pi. Let Lambda be an R-order such that Q Lambda is a separable Q-algebra. Maranda showed that there exists k is an element of N such that for all Lambda-lattices L and M, if L/L pi(k) similar or equal to M/M pi(k), then L similar or equal to M. Moreover, if R is complete and L is an indecomposable Lambda-lattice, then L/L pi(k) is also indecomposable. We extend Maranda's theorem to the class of R-reduced R-torsion-free pure-injective Lambda-modules. As an application of this extension, we show that if Lambda is an order over a Dedekind domain R with field of fractions Q such that Q Lambda is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of Lambda is isomorphic to the lattice of open subsets of the R-torsionfree part of the left Ziegler spectrum of Lambda. Furthermore, with k as in Maranda's theorem, we show that if M is R-torsion-free and H(M) is the pure-injective hull of M, then H(M)/H(M)pi(k) is the pure-injective hull of M/M pi(k). We use this result to give a characterization of R-torsion-free pure-injective A-modules and describe the pure-injective hulls of certain R-torsion-free Lambda-modules.