The Torsion Theory Generated by M-Small Modules

Abstract

Let M be a right R-module, \({\cal M}\) the class of all M-small modules, and P a projective cover of M in \(\sigma\)[M]. We consider the torsion theories \(\tau_{\cal M}\) = (\({\cal T}_{\cal M}, {\cal F}_{\cal M}\)), \(\tau_V\) = (\({\cal T}_V, {\cal F}_V\)), and \(\tau_P\) = (\({\cal T}_P, {\cal F}_P\)) in \(\sigma\)[M], where \(\tau_{\cal M}\) is the torsion theory generated by \({\cal M}, \tau_V\) is the torsion theory cogenerated by \({\cal M}\), and \(\tau_P\) is the dual Lambek torsion theory. We study some conditions for \(\tau_{\cal M}\) to be cohereditary, stable, or split, and prove that Rej(M, \({\cal M}\)) = M \(\Leftrightarrow\) \({\cal F}_P\) = \({\cal M}\) (= \({\cal T}_{\cal M}\) = \({\cal F}_V\)) \(\Leftrightarrow\) \({\cal T}_P\) = \({\cal T}_V\) \(\Leftrightarrow\) Gen M (P) \(\subseteq\) \({\cal T}_V\).

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Correspondence to A. Çiгdem Özcan.

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2000 Mathematics Subject Classification: 16S90

Communicated by Weimin Xue

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Özcan, A.Ç., Harmanci, A. The Torsion Theory Generated by M-Small Modules. Algebra Colloq. 10, 41–52 (2003). https://doi.org/10.1007/s100110300006

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Keywords:

  • hereditary torsion theory
  • small module