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COMMUTATIVE RINGS AND MODULES THAT ARE r-NOETHERIAN

  • Anebri, Adam (Laboratory of Modelling and Mathematical Structures Department of Mathematics Faculty of Science and Technology of Fez University S.M. Ben Abdellah Fez) ;
  • Mahdou, Najib (Laboratory of Modelling and Mathematical Structures Department of Mathematics Faculty of Science and Technology of Fez University S.M. Ben Abdellah Fez) ;
  • Tekir, Unsal (Department of Mathematics Marmara University)
  • Received : 2020.10.18
  • Accepted : 2021.03.10
  • Published : 2021.09.30

Abstract

In this paper, we introduce and investigate a new class of modules that is closely related to the class of Noetherian modules. Let R be a commutative ring and M be an R-module. We say that M is an r-Noetherian module if every r-submodule of M is finitely generated. Also, we call the ring R to be an r-Noetherian ring if R is an r-Noetherian R-module, or equivalently, every r-ideal of R is finitely generated. We show that many properties of Noetherian modules are also true for r-Noetherian modules. Moreover, we extend the concept of weakly Noetherian rings to the category of modules and we characterize Noetherian modules in terms of r-Noetherian and weakly Noetherian modules. Finally, we use the idealization construction to give non-trivial examples of r-Noetherian rings that are not Noetherian.

Keywords

References

  1. M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm. Algebra 36 (2008), no. 12, 4620-4642. https://doi.org/10.1080/00927870802186805
  2. D. D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16 (2017), no. 8, 1750143, 19 pp. https://doi.org/10.1142/S0219498817501432
  3. D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
  4. E. P. Armendariz, Rings with an almost Noetherian ring of fractions, Math. Scand. 41 (1977), no. 1, 15-18. https://doi.org/10.7146/math.scand.a-11699
  5. C. Bakkari, S. Kabbaj, and N. Mahdou, Trivial extensions defined by Prufer conditions, J. Pure Appl. Algebra 214 (2010), no. 1, 53-60. https://doi.org/10.1016/j.jpaa.2009.04.011
  6. A. Barnard, Multiplication modules, J. Algebra 71 (1981), no. 1, 174-178. https://doi.org/10.1016/0021-8693(81)90112-5
  7. D. E. Dobbs, A. El Khalfi, and N. Mahdou, Trivial extensions satisfying certain valuation-like properties, Comm. Algebra 47 (2019), no. 5, 2060-2077. https://doi.org/10.1080/00927872.2018.1527926
  8. T. Dumitrescu, N. Mahdou, and Y. Zahir, Radical factorization for trivial extensions and amalgamated duplication rings, J. Algebra Appl. 20 (2021), no. 2, 2150025, 10 pp. https://doi.org/10.1142/S0219498821500250
  9. Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra 16 (1988), no. 4, 755-779. https://doi.org/10.1080/00927878808823601
  10. S. Glaz, Commutative coherent rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/BFb0084570
  11. J. A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
  12. S. E. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra 32 (2004), no. 10, 3937-3953. https://doi.org/10.1081/AGB200027791
  13. S. Koc and U. Tekir, r-submodules and sr-submodules, Turkish J. Math. 42 (2018), no. 4, 1863-1876. https://doi.org/10.3906/mat-1702-20
  14. S. C. Lee, Finitely generated modules, J. Korean Math. Soc. 28 (1991), no. 1, 1-11.
  15. C. P. Lu, Spectra of modules, Comm. Algebra 23 (1995), no. 10, 3741-3752. https://doi.org/10.1080/00927879508825430
  16. N. Mahdou and A. R. Hassani, On weakly-Noetherian rings, Rend. Semin. Mat. Univ. Politec. Torino 70 (2012), no. 3, 289-296.
  17. R. Mohamadian, r-ideals in commutative rings, Turkish J. Math. 39 (2015), no. 5, 733-749. https://doi.org/10.3906/mat-1503-35
  18. M. A. Ndiaye and C. T. Gueye, On commutative EKFN-ring with ascending chain condition on annihilators, Int. J. Appl. Math. 86 (2013), 871-881.
  19. P. Ribenboim, Algebraic Numbers, Wiley-Interscience, New York, 1972.