Abstract
There is a classical result known as Baer’s Lemma that states that an R-module E is injective if it is injective for R. This means that if a map from a submodule of R, that is, from a left ideal L of R to E can always be extended to R, then a map to E from a submodule A of any R-module B can be extended to B; in other words, E is injective. In this paper, we generalize this result to the category q ω consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma is useful in proving that torsion free covers exist for q ω.
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References
R. Baer: Abelian groups that are direct summands of every containing abelian group. Bull. Amer. Math. Soc. 46 (1940), 800–806.
E. Enochs: Torsion free covering modules. Proc. Amer. Math. Soc. 14 (1963), 884–889.
M. Dunkum Wesley: Torsion free covers of graded and filtered modules. Ph.D. thesis, University of Kentucky, 2005.
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Dunkum, M. A generalization of Baer’s Lemma. Czech Math J 59, 241–247 (2009). https://doi.org/10.1007/s10587-009-0017-3
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DOI: https://doi.org/10.1007/s10587-009-0017-3