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Keywords:
Baer's Lemma; injective; representations of quivers; torsion free covers
Baer's Lemma; injective; representations of quivers; torsion free covers
Summary:
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 $q_{\omega }$ 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 $q_{\omega }$.
References:
[1] Baer, R.: Abelian groups that are direct summands of every containing abelian group. Bull. Amer. Math. Soc. 46 800-806 (1940). DOI 10.1090/S0002-9904-1940-07306-9 | MR 0002886 | Zbl 0024.14902
[2] Enochs, E.: Torsion free covering modules. Proc. Amer. Math. Soc. 14 884-889 (1963). DOI 10.1090/S0002-9939-1963-0168617-7 | MR 0168617 | Zbl 0116.26003
[3] Wesley, M. Dunkum: Torsion free covers of graded and filtered modules. Ph.D. thesis, University of Kentucky (2005). MR 2707058
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