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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ T$-faithful subcategories and localization


Author: John A. Beachy
Journal: Trans. Amer. Math. Soc. 195 (1974), 61-79
MSC: Primary 16A08
DOI: https://doi.org/10.1090/S0002-9947-1974-0364322-4
MathSciNet review: 0364322
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Abstract: For any additive functor from a category of modules into an abelian category there is a largest Giraud subcategory for which the functor acts faithfully on homomorphisms into the subcategory. It is the largest Giraud subcategory into which the functor reflects exact sequences, and under certain conditions it is just the largest Giraud subcategory on which the functor acts faithfully. If the functor is exact and has a right adjoint, then the subcategory is equivalent to the quotient category determined by the kernel of the functor. In certain cases, the construction can be applied to a Morita context in order to obtain a recent theorem of Mueller.

Similarly, the functor defines a certain reflective subcategory and an associated radical, which is a torsion radical in case the functor preserves monomorphisms. Certain results concerning this radical, when defined by an adjoint functor, can be applied to obtain two theorems of Morita on balanced modules.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0364322-4
Keywords: T-faithful subcategory, torsion radical, quotient category, balanced module
Article copyright: © Copyright 1974 American Mathematical Society