Abstract
In this article we consider injective modules relative to a torsion theory τ. We introduce τ-M-injective and s-τ-M-injective modules, relatively τ-injective modules, the τ-M-injective hull and Σ-τ-M-injective and Σ-s-τ-M-injective modules. We then examine the relationship between these new and known concepts.
Some of the new results obtained include a Generalized Fuchs Criterion characterizing s-τ-M-injective modules, a Generalized Azumaya’s Lemma characterizing τ-⊕ I -M i -injective modules, the proof of the existence and uniqueness up to isomorphism of the τ-M-injective hull and generalizations of results by Albu and Năstăsescu, Faith, and Cailleau on necessary and sufficient conditions for a module to be Σ-s-τ-M-injective, Σ-τ-injective and for a direct sum of Σ-s-τ-M-injective modules to be Σ-s-τ-M-injective.
This paper is part of the first author’s University of Otago Ph.D. thesis, written under the supervision of the second author. The first author gratefully acknowledges the support of the Commonwealth Scholarship and Fellowship Committee of New Zealand. Both authors are very grateful to the referee for their careful reading of the article, their suggestions, and for supplying reference [PRY].
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To Robert Wisbauer on the occasion of his 65th birthday.
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© 2008 Birkhäuser Verlag Basel/Switzerland
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Charalambides, S., Clark, J. (2008). τ-Injective Modules. In: Brzeziński, T., Gómez Pardo, J.L., Shestakov, I., Smith, P.F. (eds) Modules and Comodules. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8742-6_9
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DOI: https://doi.org/10.1007/978-3-7643-8742-6_9
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