Gregory, Lorna ORCID: https://orcid.org/0000-0002-5508-7217
(2023)
*Maranda’s theorem for pure-injective modules and duality.*
Canadian Journal of Mathematics, 75 (2).
pp. 581-607.
ISSN 0008-414X

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## 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.

Item Type: | Article |
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Uncontrolled Keywords: | order over a dedekind domain,ziegler spectrum,pure-injective,mathematics(all) ,/dk/atira/pure/subjectarea/asjc/2600 |

Faculty \ School: | Faculty of Science > School of Mathematics |

Related URLs: | |

Depositing User: | LivePure Connector |

Date Deposited: | 19 Oct 2022 11:30 |

Last Modified: | 17 Apr 2023 09:30 |

URI: | https://ueaeprints.uea.ac.uk/id/eprint/89201 |

DOI: | 10.4153/S0008414X22000098 |

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