Ring Characterizations with Mutually SS-Supplemented Modules

Year 2024, , 30 - 34, 24.03.2024

Burcu Nişancı Türkmen , Zehra Betül Meşeci

https://doi.org/10.17798/bitlisfen.1310501

Abstract

In this text, the notion of mutually ss-supplemented modules characterizes with the help of semiperfect rings. For this mutually ss-supplemented modules were first classified according to certain properties. These features can be listed as follows in the refinable modules, distributive modules, fully invariant submodules and (π-)projective modules. Especially it was proven that each 𝜋-projective mutually ss-supplemented module is amply mutually ss-supplemented. It was obtained that every ss-supplemented refinable module has direct summand which is a mutually ss-supplement in the module. It was shown that C=⨁_(ϱ∈Λ ) C_ϱ is a mutually ss-supplemented module which each submodule of C is a fully invariant submodule, for the family of mutually ss-supplemented modules {C_ϱ }_(ϱ∈Λ).

Keywords

Semisimple module, Mutual supplements, ss-supplement, Mutually ss-supplemented module

Rings Characterizations with Mutually SS-Supplemented Modules

Abstract

In this text, the notion of mutually ss-supplemented modules characterizes with the help of semiperfect rings. For this mutually ss-supplemented modules were first classified according to certain properties. These features can be listed as follows in the refinable modules, distributive modules, fully invariant submodules and (π-)projective modules. Especially it was proven that each 𝜋-projective mutually ss-supplemented module is amply mutually ss-supplemented. It was obtained that every ss-supplemented refinable module has direct summand which is a mutually ss-supplement in the module. It was shown that C=⨁_(ϱ∈Λ ) C_ϱ is a mutually ss-supplemented module which each submodule of C is a fully invariant submodule, for the family of mutually ss-supplemented modules {C_ϱ }_(ϱ∈Λ).

Keywords

Semisimple modules, Mutually ss-supplemented modules, Mutual supplements, ss-supplement

References

• [1] F. W. Anderson, K. R. Fuller, Graduate Texts in Mathematics. Rings and Categories of Modules, Springer-Verlag, 1992.
• [2] T. W. Hungerford, Algebra. Springer Verlag, 502, New York, 1973.
• [3] F. Kasch, Modules and Rings. Published for the London Mathematical Society by Academic Press, 372, Teubner, 1982.
• [4] E. Kaynar, E. Türkmen and H. Çalışıcı, SS-supplemented modules. Communications Faculty of Sciences. University of Ankara Series A1 Mathematics and Statistics, vol. 69, no. 1, pp. 473-485, 2020.
• [5] E. Kaynar, ⨁_ss-supplemented modules. New Trends in Rings and Modules (NTRM 2018), Gebze Technical University. Abstract Book, pp. 3 (June 2018).
• [6] B. Koşar and C. Nebiyev, “Tg-supplemented modules,” Miskolc Mathematical Notes, vol. 16, no. 2, pp. 979–986, 2015.
• [7] Z. Betül Meşeci, B. Nişancı Türkmen, Mutually SS-Supplemented Modules, 1st Iceans 2022, Proceeding Book, pp 2419-2422, 2022.
• [8] A. Ç. Özcan, A. Harmancı, P. F. Smith, “Duo modules,” Glasgow Math. J., vol. 48, pp. 533-545, 2006.
• [9] J. J. Rotman, An Introduction to Homological Algebra. Second Edition. Springer Science+Business Media, LLC, 2009.
• [10] R. Wisbauer, Foundations of Module and Ring Theory. Gordon and Breach,600, Philadelphia, 1991.
• [11] R. Wisbauer, Modules and Algebras: Bimodule Structure on Group Actions and Algebras (Vol. 81). CRC Press, 1996.