On submodule transitivity of QTAG-modules

Fahad Sikander; Firdhousi Begam; Tanveer Fatima; Fahad Sikander; Firdhousi Begam; Tanveer Fatima

doi:10.3934/math.2023467

AIMS Mathematics

2023, Volume 8, Issue 4: 9303-9313. doi: 10.3934/math.2023467

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Research article

On submodule transitivity of QTAG-modules

1.
Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Jeddah 23442, Saudi Arabia
2.
Applied Science Section, University Polytechnic, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India
3.
Department of Mathematics and Statistics, College of Sciences, Taibah University, Yanbu 41911, Saudi Arabia

Received: 10 November 2022 Revised: 27 December 2022 Accepted: 08 February 2023 Published: 15 February 2023
MSC : 20K10

In this paper, we generalize a suitable transformation from an element-based to a submodule-based interpretation of the traditional idea of transitivity in QTAG modules. We examine QTAG modules that are transitive in the sense that the module has an automorphism that sends one isotype submodule $ K $ onto any other isotype submodule $ K' $, unless this is impossible because either the submodules or the quotient modules are not isomorphic. Additionally, the classes of strongly transitive and strongly $ U $-transitive QTAG modules are defined using a slight adaptations of this. This work investigates the latter class in depth, demonstrating that every $ \alpha $- module is strongly transitive with regard to countably generated isotype submodules.
- QTAG modules,
- nice submodules,
- totally projective module,
- isotype submodule,
- Ulm-Kaplansky invariants
Citation: Fahad Sikander, Firdhousi Begam, Tanveer Fatima. On submodule transitivity of QTAG-modules[J]. AIMS Mathematics, 2023, 8(4): 9303-9313. doi: 10.3934/math.2023467

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Abstract

In this paper, we generalize a suitable transformation from an element-based to a submodule-based interpretation of the traditional idea of transitivity in QTAG modules. We examine QTAG modules that are transitive in the sense that the module has an automorphism that sends one isotype submodule $ K $ onto any other isotype submodule $ K' $, unless this is impossible because either the submodules or the quotient modules are not isomorphic. Additionally, the classes of strongly transitive and strongly $ U $-transitive QTAG modules are defined using a slight adaptations of this. This work investigates the latter class in depth, demonstrating that every $ \alpha $- module is strongly transitive with regard to countably generated isotype submodules.

References

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