대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제38권4호
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  • Pages.993-999
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  • 2023
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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ON GRADED (m, n)-CLOSED SUBMODULES

  • Rezvan Varmazyar (Department of Mathematics Khoy Branch, Islamic Azad University)
  • 투고 : 2022.11.19
  • 심사 : 2023.03.30
  • 발행 : 2023.10.31
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초록

Let A be a G-graded commutative ring with identity and M a graded A-module. Let m, n be positive integers with m > n. A proper graded submodule L of M is said to be graded (m, n)-closed if amg·xt ∈ L implies that ang·xt ∈ L, where ag ∈ h(A) and xt ∈ h(M). The aim of this paper is to explore some basic properties of these class of submodules which are a generalization of graded (m, n)-closed ideals. Also, we investigate GCmn - rad property for graded submodules.

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과제정보

The author would like to thank the referee for valuable suggestions and comments which improved the present article.

참고문헌

  1. K. Al-Zoubi, R. Abu-Dawwas, and I. Al-Ayyoub, Graded semiprime submodules and graded semi-radical of graded submodules in graded modules, Ric. Mat. 66 (2017), no. 2, 449-455. https://doi.org/10.1007/s11587-016-0312-x
  2. D. F. Anderson and A. R. Badawi, On (m, n)-closed ideals of commutative rings, J. Algebra Appl. 16 (2017), no. 1, 1750013, 21 pp. https://doi.org/10.1142/S021949881750013X
  3. S. E. Atani,On graded prime submodules, Chiang Mai J. Sci. 33 (2006), no. 1, 3-7.
  4. S. C. Lee and R. Varmazyar, Semiprime submodules of graded multiplication modules, J. Korean Math. Soc. 49 (2012), no. 2, 435-447. https://doi.org/10.4134/JKMS.2012.49.2.435
  5. S. C. Lee and R. Varmazyar, Semiprime submodules of a module and related concepts, J. Algebra Appl. 18 (2019), no. 8, 1950147, 11 pp. https://doi.org/10.1142/S0219498819501470
  6. C. Nastasescu and F. M. J. Van Oystaeyen, Graded and filtered rings and modules, Lecture Notes in Mathematics, 758, Springer, Berlin, 1979.
  7. C. Nastasescu and F. M. J. Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, 28, North-Holland, Amsterdam, 1982.
  8. A. Pekin, S. Koc, and E. A. Ugurlu, On (m, n)-semiprime submodules, Proc. Est. Acad. Sci. 70 (2021), no. 3, 260-267. https://doi.org/10.3176/proc.2021.3.05
  9. H. Saber, T. A. Alraqad, and R. Abu-Dawwas, On graded s-prime submodules, AIMS Math. 6 (2021), no. 3, 2510-2524. https://doi.org/10.3934/math.2021152
  10. B. Sarac, On semiprime submodules, Comm. Algebra 37 (2009), no. 7, 2485-2495. https://doi.org/10.1080/00927870802101994
  11. R. Varmazyar, Graded coprime submodules, Math. Sci. (Springer) 6 (2012), Art. 70, 4 pp.