Research Article
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Year 2024, Volume: 17 Issue: 1, 268 - 278, 28.03.2024

Fatih Karabacak Özgür Taşdemir

https://doi.org/10.18185/erzifbed.1371910

Abstract

Project Number

2210E164

References

  • [1] Clark J., Lomp C., Vanaja N., Wisbauer R., (2008) Lifting modules: supplements and projectivity in module theory. Springer Science & Business Media.
  • [2] Behboodi M., Daneshvar A., Vedadi M. R., (2018) Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem, Comm. Algebra, 46(6), 2384-239.
  • [3] Karabacak F., Koşan M. T., Quynh T. C., Taşdemir Ö., (2022) On modules and rings in which complements are isomorphic to direct summands. Comm. Algebra, 50(3), 1154- 1168.
  • [4] Taşdemir Ö., Karabacak F., (2019) Generalized SIP-modules, Hacettepe J. Math. Stat. 48(4), 1137-1145.
  • [5] Taşdemir Ö., Karabacak F., (2020) Generalized SSP-modules, Comm. Algebra, 48(3), 1068-1078.
  • [6] Nicholson W. K., Yousif M. F., (2003) Quasi-Frobenius Rings. Cambridge Univ. Press.
  • [7] Smith P. F., (1992) Modules for which every submodule has unique closure. Proceedings of the Biennial Ohio-Denison Conference 302-313.
  • [8] Hadi, I.M-I., Ghawi, Th.Y., (2014) Modules with the closed sum property, International Mathematical Forum, 9(32), 1539-1551.
  • [9] Khuri S. M., (1979) Endomorphism rings and lattice isomorphisms, Journal of Algebra 56(2), 401-408, 1979.
  • [10] Ghorbani A., Vedadi M. R., (2009) Epi-retractable modules and some applications, Bull. Iran. Math. Soc., 35(1), 155-166.
  • [11] Chatters A. W., Khuri S. M., (1980) Endomorphism rings of modules over non-singular CS rings. Journal of the London Mathematical Society, 2(3), 434-444.
  • [12] Rizvi S. T., Roman C. S., (2009) On direct sums of Baer modules, Journal of Algebra, 321(2), 682-696.
  • [13] Dung N. V., Huynh D. V., Smith P. F., Wisbauer R., (1994) Extending Modules, Pitman RN Mathematics, (Vol. 313). Harlow: Longman., Scientific & Technical.
  • [14] Călugăreanu G., Schultz P. (2010) Modules with Abelian endomorphism rings. Bull. Aust. Math. Soc. 82(1), 99-112.
  • [15] Özcan A. C., Harmanci A., Smith P. F., (2006) Duo modules, Glasg. Math. J. 48, 533- 545.
  • [16] Lee G., Rizvi S. T., Roman C. S., (2010) Rickart Modules. Commun. Algebra, 38(11), 4005-4027.
  • [17] Lee G., Rizvi S. T., Roman C. S., (2011) Dual Rickart modules, Commun. Algebra, 39(11), 4036-4058.
  • [18] Mohamed S. H., Müller B. J., (1990) Continuous and Discrete Modules, London Math. Soc. Lecture Note Series, 147, Cambridge: Cambridge University Press.
  • [19] Wisbauer R., (1991) Foundations of Module and Ring Theory, Gordon and Breach, Reading.
  • [20] Nicholson W. K., Campos E. S., (2005) Morphic modules, Comm. Algebra, 33(8), 2629- 2647.
  • [21] Dung N. V., (1991) Modules whose closed submodules are finitely generated. Proc. Edinb. Math. Soc. 34, 161–166.

A Research on the Generalizations of Modules Whose Submodules are Isomorphic to a Direct Summand

Year 2024, Volume: 17 Issue: 1, 268 - 278, 28.03.2024

Fatih Karabacak Özgür Taşdemir

https://doi.org/10.18185/erzifbed.1371910

Abstract

A module M is called virtually semisimple (resp. virtually extending) if every submodule (resp. complement submodule) of M is isomorphic to a direct summand of M. It is known that virtually extending modules is a generalization of virtually semisimple modules. In this paper, the relationships between virtually extending modules and other generalizations of virtually semisimple modules are examined. Moreover, we introduce a new generalization of virtually semisimple modules; namely CH modules: We say a module M is a c-epi-retractable (or briefly CH module) if any complement submodule of M is a homomorphic image of M. CH modules contains the class of virtually extending modules and the class of epi-retractable modules. We also give some basic properties of this new module class.

Keywords

virtually extending module epi-retractable module CH module virtually semisimple module

Ethical Statement

There are no ethical issues regarding the publication of this study.

Supporting Institution

Anadolu University Scientific Research Projects Commission

Project Number

2210E164

References

  • [1] Clark J., Lomp C., Vanaja N., Wisbauer R., (2008) Lifting modules: supplements and projectivity in module theory. Springer Science & Business Media.
  • [2] Behboodi M., Daneshvar A., Vedadi M. R., (2018) Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem, Comm. Algebra, 46(6), 2384-239.
  • [3] Karabacak F., Koşan M. T., Quynh T. C., Taşdemir Ö., (2022) On modules and rings in which complements are isomorphic to direct summands. Comm. Algebra, 50(3), 1154- 1168.
  • [4] Taşdemir Ö., Karabacak F., (2019) Generalized SIP-modules, Hacettepe J. Math. Stat. 48(4), 1137-1145.
  • [5] Taşdemir Ö., Karabacak F., (2020) Generalized SSP-modules, Comm. Algebra, 48(3), 1068-1078.
  • [6] Nicholson W. K., Yousif M. F., (2003) Quasi-Frobenius Rings. Cambridge Univ. Press.
  • [7] Smith P. F., (1992) Modules for which every submodule has unique closure. Proceedings of the Biennial Ohio-Denison Conference 302-313.
  • [8] Hadi, I.M-I., Ghawi, Th.Y., (2014) Modules with the closed sum property, International Mathematical Forum, 9(32), 1539-1551.
  • [9] Khuri S. M., (1979) Endomorphism rings and lattice isomorphisms, Journal of Algebra 56(2), 401-408, 1979.
  • [10] Ghorbani A., Vedadi M. R., (2009) Epi-retractable modules and some applications, Bull. Iran. Math. Soc., 35(1), 155-166.
  • [11] Chatters A. W., Khuri S. M., (1980) Endomorphism rings of modules over non-singular CS rings. Journal of the London Mathematical Society, 2(3), 434-444.
  • [12] Rizvi S. T., Roman C. S., (2009) On direct sums of Baer modules, Journal of Algebra, 321(2), 682-696.
  • [13] Dung N. V., Huynh D. V., Smith P. F., Wisbauer R., (1994) Extending Modules, Pitman RN Mathematics, (Vol. 313). Harlow: Longman., Scientific & Technical.
  • [14] Călugăreanu G., Schultz P. (2010) Modules with Abelian endomorphism rings. Bull. Aust. Math. Soc. 82(1), 99-112.
  • [15] Özcan A. C., Harmanci A., Smith P. F., (2006) Duo modules, Glasg. Math. J. 48, 533- 545.
  • [16] Lee G., Rizvi S. T., Roman C. S., (2010) Rickart Modules. Commun. Algebra, 38(11), 4005-4027.
  • [17] Lee G., Rizvi S. T., Roman C. S., (2011) Dual Rickart modules, Commun. Algebra, 39(11), 4036-4058.
  • [18] Mohamed S. H., Müller B. J., (1990) Continuous and Discrete Modules, London Math. Soc. Lecture Note Series, 147, Cambridge: Cambridge University Press.
  • [19] Wisbauer R., (1991) Foundations of Module and Ring Theory, Gordon and Breach, Reading.
  • [20] Nicholson W. K., Campos E. S., (2005) Morphic modules, Comm. Algebra, 33(8), 2629- 2647.
  • [21] Dung N. V., (1991) Modules whose closed submodules are finitely generated. Proc. Edinb. Math. Soc. 34, 161–166.
There are 21 citations in total.

Details

Primary Language English
Subjects Numerical Methods in Mechanical Engineering
Journal Section Makaleler
Authors

Fatih Karabacak 0000-0003-4925-512X

Özgür Taşdemir 0000-0003-2500-8255

Project Number 2210E164
Early Pub Date March 27, 2024
Publication Date March 28, 2024
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA Karabacak, F., & Taşdemir, Ö. (2024). A Research on the Generalizations of Modules Whose Submodules are Isomorphic to a Direct Summand. Erzincan University Journal of Science and Technology, 17(1), 268-278. https://doi.org/10.18185/erzifbed.1371910
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