Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE NILPOTENCY OF THE PRIME RADICAL OF A GOLDIE MODULE

  • John A., Beachy (Department of Mathematical Sciences Northern Illinois University) ;
  • Mauricio, Medina-Barcenas (Facultad de Ciencias Fisico-Matematicas Benemerita Universidad Autonoma de Puebla)
  • Received : 2022.01.21
  • Accepted : 2022.09.20
  • Published : 2023.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

With the notion of prime submodule defined by F. Raggi et al. we prove that the intersection of all prime submodules of a Goldie module M is a nilpotent submodule provided that M is retractable and M(Λ)-projective for every index set Λ. This extends the well known fact that in a left Goldie ring the prime radical is nilpotent.

Keywords

Acknowledgement

The second author wants to thank the members of the Facultad de Ciencias Fisico Matematicas, Benemerita Universidad Autonoma de Puebla for their hospitality during the development of this research.

References

  1. J. A. Beachy, M-injective modules and prime M-ideals, Comm. Algebra 30 (2002), no. 10, 4649-4676. https://doi.org/10.1081/AGB-120014660 
  2. J. A. Beachy and M. Medina-Barcenas, Fully prime modules and fully semiprime modules, Bull. Korean Math. Soc. 57 (2020), no. 5, 1177-1193. https://doi.org/10.4134/BKMS.b190864 
  3. J. A. Beachy and M. Medina-Barcenas, Reduced rank in σ[M], preprint, ArXiv:2201. 07196, 2022. 
  4. L. Bican, P. Jambor, T. Kepka, and P. Nemec, Prime and coprime modules, Fund. Math. 107 (1980), no. 1, 33-45. https://doi.org/10.4064/fm-107-1-33-45 
  5. A. Haghany and M. R. Vedadi, Study of semi-projective retractable modules, Algebra Colloq. 14 (2007), no. 3, 489-496. https://doi.org/10.1142/S1005386707000442 
  6. I. N. Herstein and L. Small, Nil rings satisfying certain chain conditions, Canadian J. Math. 16 (1964), 771-776. https://doi.org/10.4153/CJM-1964-074-0 
  7. T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4684-0406-7 
  8. C. Lanski, Nil subrings of Goldie rings are nilpotent, Canadian J. Math. 21 (1969), 904-907. https://doi.org/10.4153/CJM-1969-098-x 
  9. M. Medina-Barcenas and A. Ozcan, Primitive submodules, co-semisimple and regular modules, Taiwanese J. Math. 22 (2018), no. 3, 545-565. https://doi.org/10.11650/tjm/171102 
  10. M. G. Medina-Barcenas, L. A. Zaldivar-Corichi, and M. L. S. Sandoval-Miranda, A generalization of quantales with applications to modules and rings, J. Pure Appl. Algebra 220 (2016), no. 5, 1837-1857. https://doi.org/10.1016/j.jpaa.2015.10.004 
  11. J. C. Perez, M. Medina Barcenas, and J. Rios Montes, Modules with ascending chain condition on annihilators and Goldie modules, Comm. Algebra 45 (2017), no. 6, 2334-2349. https://doi.org/10.1080/00927872.2016.1233200 
  12. J. C. Perez, M. Medina Barcenas, J. Rios Montes, and A. Zaldivar Corichi, On semiprime Goldie modules, Comm. Algebra 44 (2016), no. 11, 4749-4768. https://doi.org/10.1080/00927872.2015.1113290 
  13. J. C. Perez and J. Rios Montes, Prime submodules and local Gabriel correspondence in σ[M], Comm. Algebra 40 (2012), no. 1, 213-232. https://doi.org/10.1080/00927872.2010.529095 
  14. F. Raggi, J. Rios, H. Rincon, R. Fernandez-Alonso, and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (2005), no. 4, 451-466. https://doi.org/10.1142/S0219498805001290 
  15. F. Raggi, J. Rios, H. Rincon, R. Fernandez-Alonso, and C. Signoret, Semiprime preradicals, Comm. Algebra 37 (2009), no. 8, 2811-2822. https://doi.org/10.1080/00927870802623476 
  16. J. E. van den Berg and R. Wisbauer, Modules whose hereditary pretorsion classes are closed under products, J. Pure Appl. Algebra 209 (2007), no. 1, 215-221. https://doi.org/10.1016/j.jpaa.2006.05.030 
  17. R. Wisbauer, Foundations of module and ring theory, revised and translated from the 1988 German edition, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.