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Published online by Cambridge University Press: 17 March 2022
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
$k\in \mathbb {N}$ such that for all
$\Lambda $-lattices L and M, if
$L/L\pi ^k\simeq M/M\pi ^k$, then
$L\simeq M$. 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-torsion-free 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
$\Lambda $-modules and describe the pure-injective hulls of certain R-torsion-free
$\Lambda $-modules.
The majority of this work was completed while the author was employed by the University of Camerino.