The Osofsky-Smith Theorem for Modular Lattices, and Applications (II)

Abstract

This is the second part of the paper with the same title published in Communications in Algebra in 2011. It contains applications of the Latticial Osofsky–Smith Theorem to Grothendieck categories and module categories equipped with a torsion theory. Various many different meanings spread in the literature of the relative concepts with respect to a hereditary torsion theory τ on Mod-R like τ-essential submodule, τ-complement submodule, τ-CS module, etc. are also discussed.

Key Words:

• Finitely generated object
• Grothendieck category
• Locally finitely generated category
• The Categorical Osofsky-Smith Theorem
• The Latticial Osofsky-Smith Theorem
• The Relative Osofsky–Smith Theorem
• Torsion theory
• τ-CS module
• τ-Compact module
• τ-Compactly generated module
• τ-Finitely generated module

2010 Mathematics Subject Classification:

• 06C05
• 06B35
• 16S90
• 18E15

ACKNOWLEDGMENTS

The author thanks N. V. Dung for drawing to his attention the paper Dung [19], Constantin Năstăsescu and John Van Den Berg for helpful discussions, Septimiu Crivei for providing him references [13, 14, 16], and the referee for his/her careful reading of the manuscript and helpful suggestions.

Notes

Communicated by J. L. Gomez Pardo.