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Direct Sums of Strongly Lifting Modules

  • Received : 2019.06.20
  • Accepted : 2020.08.18
  • Published : 2020.12.31

Abstract

For the recently defined notion of strongly lifting modules, it has been shown that a direct sum is not, in general, strongly lifting. In this paper we investigate the question: When are the direct sums of strongly lifting modules, also strongly lifting? We introduce the notion of a relatively strongly projective module and use it to show if M = M1 ⊕ M2 is amply supplemented, then M is strongly lifting if and only if M1 and M2 are relatively strongly projective and strongly lifting. Also, we consider when an arbitrary direct sum of hollow (resp. local) modules is strongly lifting.

Keywords

Acknowledgement

We would like to thank the referees for valuable comments.

References

  1. T. Amouzegar, D. K. Tutuncu and Y. Talebi, t-dual Baer modules and t-lifting modules, Vietnam J. Math., 42(2)(2014), 159-169. https://doi.org/10.1007/s10013-013-0045-z
  2. G. F. Birkenmeier, B. J. Muller and S. T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Comm. Algebra, 30(2002), 1395-1415. https://doi.org/10.1080/00927870209342387
  3. G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Modules with fully invariant submodules essential in fully invariant summands, Comm. Algebra, 30(2002), 1833-1852. https://doi.org/10.1081/AGB-120013220
  4. J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, supplements and projectivity in module theory, Frontiers in Mathematics, Birkhauser Verlag, 2006.
  5. S. Ebrahimi Atani, M. Khoramdel and S. Dolati Pish Hesari, Strongly extending modules, Kyungpook Math. J., 54(2014), 237-247. https://doi.org/10.5666/kmj.2014.54.2.237
  6. S. Ebrahimi Atani, M. Khoramdel and S. Dolati Pish Hesari, t-dual Rickart modules, Bull. Iranian Math. Soc., 42(3)(2016), 627-642.
  7. S. Ebrahimi Atani, M. Khoramdel and S. Dolati Pish Hesari, C3-modules, Demonstr. Math., 49(3)(2016), 282-292. https://doi.org/10.1515/dema-2016-0024
  8. M. Khoramdel, On Baer and Rickart modules and some results on semirings, Ph.D thesis, University of Guilan, 2015.
  9. G. Lee, S. T. Rizvi, and C. S. Roman, Dual Rickart modules, Comm. Algebra, 39(11)(2011), 4036-4058. https://doi.org/10.1080/00927872.2010.515639
  10. S. H. Mohamed and B. J. Muller, Continuous and discrete modules, London Mathematical Society Lecture Note Series 147, Cambridge University Press, Cambridge, 1990.
  11. T. Takeuchi, On cofinite-dimensional modules, Hokkaido Math J., 5(1976), 1-43. https://doi.org/10.14492/hokmj/1381758746
  12. Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra, 30(2002), 1449-1460. https://doi.org/10.1080/00927870209342390
  13. D. K. Tutuncu, On lifting modules, Comm. Algebra, 28(2000), 3427-3440. https://doi.org/10.1080/00927870008827034
  14. D. K. Tutuncu and R. Tribak, On dual Baer modules, Glasg. Math. J., 52(2010), 261-269. https://doi.org/10.1017/S0017089509990334
  15. Y. Wang, Strongly lifting modules and strongly dual Rickart modules, Front. Math. China, 12(1)(2017), 219-229. https://doi.org/10.1007/s11464-016-0599-7
  16. R. Wisbauer, Foundations of module and ring theory, Gordon and Breach Science Publishers, Philadelphia, 1991.