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Graded S-1-absorbing prime submodules in graded multiplication modules

Year 2022, Volume: 32 Issue: 32, 62 - 79, 16.07.2022
https://doi.org/10.24330/ieja.1081701

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a commutative $G$-graded ring with non-zero identity, $S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ a graded $R$-module. In this article, we introduce and study the concept of graded $S$-1-absorbing prime submodules. A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-1-absorbing prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then either $s_{g}a_{h}b_{h'}\in (N:_{R}M)$ or $s_{g}m_{k}\in N$ for all non-unit elements $a_{h},b_{h'}\in h(R)$ and all $m_{k}\in h(M)$. Some examples, characterizations and properties of graded $S$-1-absorbing prime submodules are given. Moreover, we give some characterizations of graded $S$-1-absorbing prime submodules in graded multiplicative modules.

References

  • R. Abu-Dawwas, M. Bataineh and M. Al-Muanger, Graded prime submodules over non-commutative rings, Vietnam J. Math., 46(3) (2018), 681-692.
  • K. Al-Zoubi and R. Abu-Dawwas, On graded 2-absorbing and weakly graded 2-absorbing submodules, Journal of Mathematical Sciences: Advances and Applications, 28 (2014), 45-60.
  • S. Ebrahimi Atani and F. Farzalipour, Notes on the graded prime submodules, Int. Math. Forum, (1) (2006), 1871-1880.
  • S. Ebrahimi Atani and F. Farzalipour, On graded secondary modules, Turkish J. Math., 31 (2007), 371-378.
  • J. Escoriza and B. Torrecillas, Multiplication objects in commutative Grothendieck categories, Comm. Algebra, 26(6) (1998), 1867-1883.
  • F. Farzalipour and P. Ghiasvand, On $S$-1-absorbing prime submodules, J. Algebra Appl., to appear.
  • F. Farzalipour and P. Ghiasvand, On the union of graded prime submodules, Thai. J. Math., 9(1) (2011), 49-55.
  • F. Farzalipour and P. Ghiasvand, On graded weak multiplication modules, Tamkang J. Math., 43(2) (2012), 171-177.
  • F. Farzalipour, P. Ghiasvand and M. Adlifard, On graded weakly semiprime submodules, Thai. J. Math., 12(1) (2014), 167-174.
  • P. Ghiasvand and F. Farzalipour, Some properties of graded multiplication modules, Far East J. Math. Sci., 34(3) (2009), 341-352.
  • P. Ghiasvand and F. Farzalipour, On the graded primary radical of graded submodules, Adv. Appl. Math. Sci., 10(1) (2011), 1-7.
  • K. H. Oral, Ü. Tekir and A. G. Ağargün, On graded prime and primary submodules, Turkish J. Math., 35 (2011), 159-167.
  • N. Nastasescu and F. van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1982.
  • N. Nastasescu and F. van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, vol. 1836., Springer-Verlag, Berlin, 2004.
Year 2022, Volume: 32 Issue: 32, 62 - 79, 16.07.2022
https://doi.org/10.24330/ieja.1081701

Abstract

References

  • R. Abu-Dawwas, M. Bataineh and M. Al-Muanger, Graded prime submodules over non-commutative rings, Vietnam J. Math., 46(3) (2018), 681-692.
  • K. Al-Zoubi and R. Abu-Dawwas, On graded 2-absorbing and weakly graded 2-absorbing submodules, Journal of Mathematical Sciences: Advances and Applications, 28 (2014), 45-60.
  • S. Ebrahimi Atani and F. Farzalipour, Notes on the graded prime submodules, Int. Math. Forum, (1) (2006), 1871-1880.
  • S. Ebrahimi Atani and F. Farzalipour, On graded secondary modules, Turkish J. Math., 31 (2007), 371-378.
  • J. Escoriza and B. Torrecillas, Multiplication objects in commutative Grothendieck categories, Comm. Algebra, 26(6) (1998), 1867-1883.
  • F. Farzalipour and P. Ghiasvand, On $S$-1-absorbing prime submodules, J. Algebra Appl., to appear.
  • F. Farzalipour and P. Ghiasvand, On the union of graded prime submodules, Thai. J. Math., 9(1) (2011), 49-55.
  • F. Farzalipour and P. Ghiasvand, On graded weak multiplication modules, Tamkang J. Math., 43(2) (2012), 171-177.
  • F. Farzalipour, P. Ghiasvand and M. Adlifard, On graded weakly semiprime submodules, Thai. J. Math., 12(1) (2014), 167-174.
  • P. Ghiasvand and F. Farzalipour, Some properties of graded multiplication modules, Far East J. Math. Sci., 34(3) (2009), 341-352.
  • P. Ghiasvand and F. Farzalipour, On the graded primary radical of graded submodules, Adv. Appl. Math. Sci., 10(1) (2011), 1-7.
  • K. H. Oral, Ü. Tekir and A. G. Ağargün, On graded prime and primary submodules, Turkish J. Math., 35 (2011), 159-167.
  • N. Nastasescu and F. van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1982.
  • N. Nastasescu and F. van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, vol. 1836., Springer-Verlag, Berlin, 2004.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Farkhondeh Farzalıpour This is me

Peyman Ghıasvand This is me

Publication Date July 16, 2022
Published in Issue Year 2022 Volume: 32 Issue: 32

Cite

APA Farzalıpour, F., & Ghıasvand, P. (2022). Graded S-1-absorbing prime submodules in graded multiplication modules. International Electronic Journal of Algebra, 32(32), 62-79. https://doi.org/10.24330/ieja.1081701
AMA Farzalıpour F, Ghıasvand P. Graded S-1-absorbing prime submodules in graded multiplication modules. IEJA. July 2022;32(32):62-79. doi:10.24330/ieja.1081701
Chicago Farzalıpour, Farkhondeh, and Peyman Ghıasvand. “Graded S-1-Absorbing Prime Submodules in Graded Multiplication Modules”. International Electronic Journal of Algebra 32, no. 32 (July 2022): 62-79. https://doi.org/10.24330/ieja.1081701.
EndNote Farzalıpour F, Ghıasvand P (July 1, 2022) Graded S-1-absorbing prime submodules in graded multiplication modules. International Electronic Journal of Algebra 32 32 62–79.
IEEE F. Farzalıpour and P. Ghıasvand, “Graded S-1-absorbing prime submodules in graded multiplication modules”, IEJA, vol. 32, no. 32, pp. 62–79, 2022, doi: 10.24330/ieja.1081701.
ISNAD Farzalıpour, Farkhondeh - Ghıasvand, Peyman. “Graded S-1-Absorbing Prime Submodules in Graded Multiplication Modules”. International Electronic Journal of Algebra 32/32 (July 2022), 62-79. https://doi.org/10.24330/ieja.1081701.
JAMA Farzalıpour F, Ghıasvand P. Graded S-1-absorbing prime submodules in graded multiplication modules. IEJA. 2022;32:62–79.
MLA Farzalıpour, Farkhondeh and Peyman Ghıasvand. “Graded S-1-Absorbing Prime Submodules in Graded Multiplication Modules”. International Electronic Journal of Algebra, vol. 32, no. 32, 2022, pp. 62-79, doi:10.24330/ieja.1081701.
Vancouver Farzalıpour F, Ghıasvand P. Graded S-1-absorbing prime submodules in graded multiplication modules. IEJA. 2022;32(32):62-79.

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