H-supplemented modules with respect to images of fully invariant submodules
DOI:
https://doi.org/10.22199/issn.0717-6279-2021-01-0003Keywords:
H-supplemented module, IF -lifting module, IF-H-supplemented module, Dual Rickart module, Endomorphisms ringAbstract
Lifting modules plays important roles in module theory. H-supplemented modules are a nice generalization of lifting modules which have been studied extensively recently. In this article, we introduce a proper generalization of H-supplemented modules via images of fully invariant submodules. Let F be a fully invariant submodule of a right Rmodule M. We say that M is IF -H-supplemented in case for every endomorphism φ of M, there is a direct summand D of M such that φ(F) + X = M if and only if D + X = M, for every submodule X of M. It is proved that M is IF -H-supplemented if and only if F is a dual Rickart direct summand of M for a fully invariant noncosingular submodule F of M. It is shown that the direct sum of IF –H supplemented modules is not in general IF -H-supplemented. Some sufficient conditions such that the direct sum of IF -H-supplemented modules is IF -H-supplemented are given
References
T. Amouzegar, “A Generalization of Lifting Modules”, Ukrainian mathematical journal, vol. 66, no. 11, pp. 1654–1664, Apr. 2015, doi: 10.1007/s11253-015-1042-z
T. Amouzegar and A. R. Moniri Hamzekolaee, Lifting modules with respect to images of a fully invariant submodule, Novi Sad journal mathematical, doi: 10.30755/NSJOM.09413
G. F. Birkenmeier, F.T. Mutlu, C. Nebiyev, N. Sokmez and A. Tercan, “Goldie∗-supplemented modules”, Glasgow mathematics journal, vol. 52, no. A, pp. 41-52, Jun. 2010, doi: 10.1017/S0017089510000212
J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting modules supplements and projectivy in module theory. Basel: Birkhäuser, 2006, doi: 10.1007/3-7643-7573-6
A. Ghorbani and M. R. Vedadi, “Epi-retractable modules and some applications”, Bulletin Iranian Mathematics Society, vol. 35, no. 1, pp. 155-166, 2009. [On line]. Available: https://bit.ly/3hiLmRO
D. Keskin, “Finite direct sums of D1-modules”, Turkish journal mathematical, vol. 22, pp. 85-91, 1998. [On line]. Available: https://bit.ly/2KxT5zA
D. Keskin, M. J. Nematollahi, and Y. Talebi, “On H-Supplemented Modules”, Algebra colloquium, vol. 18, no. s01, pp. 915–924, 2011, doi: 10.1142/S1005386711000794
G. Lee, S. T. Rizvi and C. S. Roman, “Dual Rickart modules”, Communications in algebra, vol. 39, no. 11, pp. 4036-4058, Nov. 2011, doi: 10.1080/00927872.2010.515639
S. H. Mohamed and B. J. Müller, Continuous and discrete modules, Cambridge: Cambridge University Press, 1990, doi: 10.1017/CBO9780511600692
A. R. Moniri Hamzekolaee, Y. Talebi, A. Harmanci and B. Ungor, “A new approach to H-supplemented modules via homomorphisms”, Turkish journal mathematical, vol. 43, pp. 1941-1955, Jul. 2018, doi: 10.3906/mat-1709-74
A. C. Özcan, A. Harmanci, and P. F. Smith, “Duo modules”, Glasgow mathematics journal, vol. 48, no. 3, pp. 533-545, Sep. 2006, doi: 10.1017/S0017089506003260
Y. Talebi and N. Vanaja, “The torsion theory cogenerated by M-small modules”, Communications in algebra, vol. 30, no. 3, pp. 1449-1460, 2002, doi: 10.1080/00927870209342390
Y. Talebi, R. Tribak, and A. R. Moniri Hamzekolaee, “On H-cofinitely supplemented modules”, Bulletin Iranian Mathematical Society, vol. 39, no. 2, pp. 325-346, 2013. [On line]. Available: https://bit.ly/2KIiCG5
R. Wisbauer, Foundations of module and ring theory, Reading: Gordon and Breach, 1991. [On line]. Available: https://bit.ly/37OJY6d
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