## Abstract

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/Lπ
^{k} ≃ M/Mπ
^{k}, then L ≃ M. Moreover, if R is complete and L is an indecomposable Λ-lattice, then L/Lπ
^{k} is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective Λ-modules. As an application of this extension, we show that if Λ is 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-torsionfree part of the left Ziegler spectrum of Λ. 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)π
^{k} is the pure-injective hull of M/Mπ
^{k}. We use this result to give a characterization of R-torsion-free pure-injective Λ-modules and describe the pure-injective hulls of certain R-torsion-free Λ-modules.

Original language | English |
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Pages (from-to) | 581-607 |

Number of pages | 27 |

Journal | Canadian Journal Of Mathematics-Journal Canadien De Mathematiques |

Volume | 75 |

Issue number | 2 |

Early online date | 17 Mar 2022 |

DOIs | |

Publication status | Published - Apr 2023 |

## Keywords

- Order over a Dedekind domain
- Ziegler spectrum
- pure-injective