On generalizations of graded multiplication modules
Abstract
Let $G$ be a group with identity $e$, $R$ be a $G$-graded ring with unity $1$ and $M$ be a $G$-graded $R$-module. In this article, we introduce the concept of graded quasi multiplication modules, where graded $M$ is said to be graded quasi multiplication if for every graded weakly prime $R$-submodule $N$ of $M$, $N=IM$ for some graded ideal $I$ of $R$. Also, we introduce the concept of graded absorbing multiplication modules; $M$ is said to be graded absorbing multiplication if $M$ has no graded $2$-absorbing $R$-submodules or for every graded $2$-absorbing $R$-submodule $N$ of $M$, $N=IM$ for some graded ideal $I$ of $R$. We prove many results concerning graded weakly prime submodules and graded $2$-absorbing submodules that will be useful in providing several properties of the two classes of graded quasi multiplication modules and graded absorbing multiplication modules.
Downloads
References
R. Abu-Dawwas, Multiplication Components of graded modules, Italian Journal of Pure and Applied Mathematics, 35 (2015), 389-392.
R. Abu-Dawwas, Graded semiprime multiplication modules, Accepted in Boletim Sociedade Paranaense de Matematica.
R. Abu-Dawwas, K. Al-Zoubi, M. Bataineh, Prime submodules of graded modules, Proyecciones Journal of Mathematics, 31 (4) (2012), 355-361. https://doi.org/10.4067/S0716-09172012000400004
R. Abu-Dawwas, M. Refai, Further results on graded prime submodules, International Journal of Algebra, 4 (28)(2010), 1413-1419.
R. Abu-Dawwas, M. Refai, Some remarks on graded weak multiplication modules, Journal of Contemporary Mathematical Sciences, 6 (14) (2011), 681-686.
R. Abu-Dawwas, M. Refai, Some remarks on Re-multiplication modules over first strongly graded rings, East-West Journal of Mathematics, 13 (1) (2011), 57-61.
K. Al-Zoubi, R. Abu-Dawwas, On graded 2-absorbing and weakly graded 2-absorbing submodules, Journal of Mathematical Sciences: Advances and Applications, 28 (2014), 45-60.
K. Al-Zoubi, R. Abu-Dawwas and S. C¸ eken, On graded 2-absorbing and graded weakly 2-absorbing ideals, Hacettepe Journal of Mathematics and Statistics, 48 (3) (2019), 724-731. https://doi.org/10.15672/HJMS.2018.543
K. Al-Zoubi, F. Qarqaz, An intersection condition for graded prime ideals, Bollettino dell'Unione Matematica Italiana, (2017). https://doi.org/10.1007/s40574-017-0148-7
S. E. Atani, On graded prime submodules, Chiang Mai Journal of Science, 33 (1)(2006), 3-7.
S. E. Atani, On graded weakly prime submodules, International Mathematical Forum, 1 (2) (2006), 61-66. https://doi.org/10.12988/imf.2006.06007
J. Escoriza, B. Torrecillas, Multiplication objects in commutative Grothendieck category, Communications in Algebra, 26 (1998), 1867-1883. https://doi.org/10.1080/00927879808826244
F. Farzalipour, P. Ghiasvand, On the union of graded prime submodules, Thai Journal of Mathematics, 9 (1) (2011), 49-55. https://doi.org/10.5402/2011/939687
F. Farzalipour, P. Ghiasvand, On graded weak multiplication modules, Tamkang Journal of Mathematics, 43 (2) (2012), 171-177. https://doi.org/10.5556/j.tkjm.43.2012.712
F. Farzalipour, P. Ghiasvand, On graded semiprime submodules, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 6 (8)(2012), 1169-1172.
M. Hamoda, A. E. Ashour, On graded n-ansorbing submodules, Le Matematiche, doi:10.4418/2015.70.2.16 (2015).
K. Khaksari, F. R. Jahromi, Multiplication graded modules, International Journal of Algebra, 7 (1) (2013), 17-24. https://doi.org/10.12988/ija.2013.13003
S. C. Lee, R. Varmazyar, Semiprime submodules of graded multiplication modules, Journal of Korean Mathematical Society, 49 (2) (2012), 435-447. https://doi.org/10.4134/JKMS.2012.49.2.435
C. Nastasescu, V. F. Oystaeyen, Methods of Graded Rings, LNM 1836, Berlin-Heidelberg: Springer-Verlag (2004). https://doi.org/10.1007/b94904
K. H. Oral, U. Tekir and A. G. Agargun, On graded prime and primary submodules, Turkish Journal of Mathematics, 35 (2) (2011), 159-167.
R. N. Uregen, U. Tekir, K. P. Shum and S. Koc, On Graded 2-Absorbing Quasi Pr ¨ imary Ideals, Southeast Asian Bulletin of Mathematics, 43 (4) (2019), 601-613.
Copyright (c) 2022 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
