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On -Quasi-Semiprime Submodules

Year 2023, Volume: 22, 359 - 363, 01.09.2023

Khaldoun Al-zoubi , Shatha Alghueırı

https://doi.org/10.55549/epstem.1351023

Abstract

Let G be a group. A ring R is called a graded ring (or G-graded ring) if there exist additive subgroups Rα of R indexed by the elements α∈G such that R=⊕α∈GRαand RαRβ⊆Rαβ for all α, β∈G. If an element of R belongs to h(R)=∪α∈GRα, then it is called homogeneous. A Left R-module M is said to be a graded R-module if there exists a family of additive subgroups {Mα} α∈G of M such that M=⊕α∈GMα and RαMβ⊆Mαβ or all α,β∈G. Also if an element of M belongs to ∪α∈GMα=h(M), then it is called homogeneous. A submodule N of M is said to be a graded submodule of M if N=⊕α∈GN∩Mα:=⊕α∈GNα. Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. A proper graded submodule S of M is said to be a gr-semiprime submodule if whenever rⁿm∈S where r∈h(R), m∈h(M) and n∈Z⁺, then rm∈S. In this work, we introduce the concept of gr-quasi-semiprime submodule as a generalization of gr-semiprime submodule and give some basic properties of these classes of graded submodules. We say that a proper graded submodule S of M is a gr-quasi-semiprime submodule if (S:RM) is a gr-semiprime ideal of R.

Keywords

Graded quasi-semiprime submodule Graded semiprime submodule Graded prime

References

  • Al-Zoubi, K., Abu-Dawwas, R., & Al-Ayyoub, I. (2017). Graded semiprime submodules and graded semi-radical of graded submodules in graded modules. Ricerche di Matematica, 66(2), 449–455.
  • Atani, S.E., & Saraei, F.E.K. (2010). Graded modules which satisfy the gr-radical formula. Thai Journal of Mathematics, 8(1), 161–170.
  • Atani, S.E. (2006). On graded prime submodules. Chiang Mai Journal of Science, 33(1), 3–7.
  • Farzalipour, F., & Ghiasvand, F. P. (2013). On graded semiprime and graded weakly semiprime ideals. International Electronic Journal of Algebra, 13, 15–22.
  • Farzalipour, F., & Ghiasvand, F. P. (2012). On graded semiprime submodules. Word Academy Science Eng. Technology, 68, 694–697.
  • Hazrat, R. (2016). Graded rings and graded grothendieck groups. Cambridge: Cambridge University Press.
  • Lee, S.C. & Varmazyar, R. (2012). Semiprime submodules of graded multiplication modules. Journal of the Korean Mathematical Society, 49(2), 435–447.
  • Nastasescu, C., & Van Oystaeyen, F. (1982). Graded and filtered rings and modules, lecture notes in mathematics (p.758), New York, NY: Springer.
  • Nastasescu, C., & Van Oystaeyen, F. (1982). Graded ring theory (p.28), Amsterdam: North-Holland.
  • Nastasescu, C., & Van Oystaeyen, F. (2004). Methods of graded rings. Berlin-Heidelberg: Springer-Verlag.
  • Refai, M., & Al-Zoubi, K. (2004). On graded primary ideals. Turkish Journal of Mathematics, 28, 217–229.

Year 2023, Volume: 22, 359 - 363, 01.09.2023

Khaldoun Al-zoubi , Shatha Alghueırı

https://doi.org/10.55549/epstem.1351023

Abstract

References

  • Al-Zoubi, K., Abu-Dawwas, R., & Al-Ayyoub, I. (2017). Graded semiprime submodules and graded semi-radical of graded submodules in graded modules. Ricerche di Matematica, 66(2), 449–455.
  • Atani, S.E., & Saraei, F.E.K. (2010). Graded modules which satisfy the gr-radical formula. Thai Journal of Mathematics, 8(1), 161–170.
  • Atani, S.E. (2006). On graded prime submodules. Chiang Mai Journal of Science, 33(1), 3–7.
  • Farzalipour, F., & Ghiasvand, F. P. (2013). On graded semiprime and graded weakly semiprime ideals. International Electronic Journal of Algebra, 13, 15–22.
  • Farzalipour, F., & Ghiasvand, F. P. (2012). On graded semiprime submodules. Word Academy Science Eng. Technology, 68, 694–697.
  • Hazrat, R. (2016). Graded rings and graded grothendieck groups. Cambridge: Cambridge University Press.
  • Lee, S.C. & Varmazyar, R. (2012). Semiprime submodules of graded multiplication modules. Journal of the Korean Mathematical Society, 49(2), 435–447.
  • Nastasescu, C., & Van Oystaeyen, F. (1982). Graded and filtered rings and modules, lecture notes in mathematics (p.758), New York, NY: Springer.
  • Nastasescu, C., & Van Oystaeyen, F. (1982). Graded ring theory (p.28), Amsterdam: North-Holland.
  • Nastasescu, C., & Van Oystaeyen, F. (2004). Methods of graded rings. Berlin-Heidelberg: Springer-Verlag.
  • Refai, M., & Al-Zoubi, K. (2004). On graded primary ideals. Turkish Journal of Mathematics, 28, 217–229.
There are 11 citations in total.

Details

Primary Language English
Subjects Statistics (Other)
Journal Section Articles
Authors

Khaldoun Al-zoubi

Shatha Alghueırı

Early Pub Date August 28, 2023
Publication Date September 1, 2023
Published in Issue Year 2023 Volume: 22

Cite

APA Al-zoubi, K., & Alghueırı, S. (2023). On -Quasi-Semiprime Submodules. The Eurasia Proceedings of Science Technology Engineering and Mathematics, 22, 359-363. https://doi.org/10.55549/epstem.1351023
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