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Torsionfree precovers. (English) Zbl 1074.16002

Eigenthaler, G. (ed.) et al., Proceedings of the 66th workshop on general algebra “66. Arbeitstagung Allgemeine Algebra”, Klagenfurt, Austria, June 19–22, 2003. Klagenfurt: Verlag Johannes Heyn (ISBN 3-7084-0121-2/pbk). Contributions to General Algebra 15, 1-6 (2004).
Let \(R\) be an associative ring with an identity element and let \(\mathcal G\) be an abstract class of \(R\)-modules. A homomorphism \(\varphi\colon G\to M\) with \(G\in{\mathcal G}\) is called a \(\mathcal G\)-precover of the module \(M\), if for each homomorphism \(f\colon F\to M\), \(F\in{\mathcal G}\), there exists a homomorphism \(g\colon F\to G\) such that \(\varphi g=f\). The author shows that the existence of an \(\mathcal F\)-precover for any module, where \(\mathcal F\) is a torsionfree class associated to a hereditary torsion theory \(\sigma\), is equivalent to the existence of an \(\mathcal F\)-precover for any (relative) injective module. Moreover the existence of \(\mathcal F\)-precovers is related with the existence of precovers for other classes of modules, relative injective \(\sigma\)-torsionfree and relative exact modules, i.e. \(\sigma\)-torsionfree modules with the property that any \(\sigma\)-torsionfree homomorphic image is \(\sigma\)-injective.
For the entire collection see [Zbl 1050.08001].

MSC:

16D90 Module categories in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
18E40 Torsion theories, radicals
16D50 Injective modules, self-injective associative rings
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