Rings Whose Certain Modules are Dual Self-CS-Baer

Year 2024, Volume: 12 Issue: 3, 113 - 118

Nuray Eroğlu

https://doi.org/10.36753/mathenot.1461857

Abstract

In this work, we characterize some rings in terms of dual self-CS-Baer modules (briefly, ds-CS-Baer modules). We prove that any ring $R$ is a left and right artinian serial ring with $J^2(R)=0$ iff $R\oplus M$ is ds-CS-Baer for every right $R$-module $M$. If $R$ is a commutative ring, then we prove that $R$ is an artinian serial ring iff $R$ is perfect and every $R$-module is a direct sum of ds-CS-Baer $R$-modules. Also, we show that $R$ is a right perfect ring iff all countably generated free right $R$-modules are ds-CS-Baer.

Keywords

Dual self-CS-Baer module, Harada ring, Lifting module, Perfect ring, QF-ring, Serial ring

References

• [1] Clark, J., Lomp, C., Vanaja, N., Wisbauer, R.: Lifting Modules: Supplements and Projectivity in Module Theory. Frontiers in Mathematics, Birkhäuser (2006).
• [2] Mohamed, S. H., Müller, B. J.: Continuous and Discrete Modules. London Mathematical Society Lecture Note Series, Vol. 147, Cambridge University Press (1990).
• [3] Crivei, S., Keskin Tütüncü, D., Radu, S. M., Tribak, R.: CS-Baer and dual CS-Baer objects in abelian categories. Journal of Algebra and Its Applications. 22(10), 2350220 (2023).
• [4] Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules. 2nd edition, Springer-Verlag, New York (1992).
• [5] Crivei, S., Radu, S. M.: CS-Rickart and dual CS-Rickart objects in abelian categories. Bulletin of Belgian Mathematical Society-Simon Stevin. 29(1), 99–122 (2022).
• [6] Tribak, R.: Dual CS-Rickart modules over Dedekind domains. Algebras and Representation Theory. 23, 229–250 (2020).
• [7] Keskin, D., Smith, P. F., Xue,W.: Rings whose modules are ⊕-supplemented. Journal of Algebra. 218(2), 470–487 (1999).
• [8] Büyükaşık, E., Lomp, C.: On recent generalization of semiperfect rings. Bulletin of the Australian Mathematical Society. 78(2), 317–325 (2008).
• [9] Warfield, R. B.: Serial rings and finitely presented modules. Journal of Algebra. 37(2), 187–222 (1975).
• [10] Brandal, W.: Commutative Rings Whose Finitely Generated Modules Decompose. Lecture Notes in Mathematics, Vol. 723, Springer-Verlag, Berlin (1979).
• [11] Harmancı, A., Keskin, D., Smith, P. F.: On ⊕-supplemented modules. Acta Mathematica Hungarica. 83 , 161–169 (1999).