Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

BÉZOUT RINGS AND WEAKLY BÉZOUT RINGS

  • El Alaoui, Haitham (Sidi Mohamed Ben Abdellah University Faculty of Sciences Dhar Al Mahraz Laboratory of Geometric and Arithmetic Agebra)
  • Received : 2021.07.04
  • Accepted : 2021.12.13
  • Published : 2022.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study some properties of Bézout and weakly Bézout rings. Then, we investigate the transfer of these notions to trivial ring extensions and amalgamated algebras along an ideal. Also, in the context of domains we show that the amalgamated is a Bézout ring if and only if it is a weakly Bézout ring. All along the paper, we put the new results to enrich the current literature with new families of examples of non-Bézout weakly Bézout rings.

Keywords

Acknowledgement

The author would like to express their sincere thanks for the referee for his/her helpful suggestions and comments, which have greatly improved this paper.

References

  1. K. Adarbeh and S. Kabbaj, Matlis' semi-regularity in trivial ring extensions of integral domains, Colloq. Math. 150 (2017), no. 2, 229-241. https://doi.org/10.4064/cm7032-12-2016
  2. K. Alaoui Ismaili and N. Mahdou, Coherence in amalgamated algebra along an ideal, Bull. Iranian Math. Soc. 41 (2015), no. 3, 625-632.
  3. D. F. Anderson and A. Badawi, On φ-Prufer rings and φ-Bezout rings, Houston J. Math. 30 (2004), no. 2, 331-343.
  4. 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
  5. D. D. Anderson and M. Zafrullah, Almost Bezout domains, J. Algebra 142 (1991), no. 2, 285-309. https://doi.org/10.1016/0021-8693(91)90309-V
  6. A. Azizi and A. Nikseresht, Simplified radical formula in modules, Houston J. Math. 38 (2012), no. 2, 333-344.
  7. F. Cheniour, On B'ezout rings, Int. J. Algebra 6 (2012), no. 29-32, 1507-1511.
  8. M. D'Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal, in Commutative algebra and its applications, 155-172, Walter de Gruyter, Berlin, 2009.
  9. M. D'Anna, C. A. Finocchiaro, and M. Fontana, Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (2010), no. 9, 1633-1641. https://doi.org/10.1016/j.jpaa.2009.12.008
  10. M. D'Anna, C. A. Finocchiaro, and M. Fontana, New algebraic properties of an amalgamated algebra along an ideal, Comm. Algebra 44 (2016), no. 5, 1836-1851. https://doi.org/10.1080/00927872.2015.1033628
  11. M. D'Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2007), no. 2, 241-252. https://doi.org/10.1007/s11512-006-0038-1
  12. M. D'Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (2007), no. 3, 443-459. https://doi.org/10.1142/S0219498807002326
  13. H. El Alaoui and H. Mouanis, About weakly B'ezout rings, Bol. Soc. Paran. Mat. 40 (2022). https://doi.org/10.5269/bspm.44605
  14. L. Gillman and M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956), 362-365. https://doi.org/10.2307/1993053
  15. S. Glaz, Commutative coherent rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/BFb0084570
  16. J. A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
  17. S. Kabbaj, On the dimension theory of polynomial rings over pullbacks, in Multiplicative ideal theory in commutative algebra, 263-277, Springer, New York, 2006. https://doi.org/10.1007/978-0-387-36717-0_16
  18. S. 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
  19. M. Kabbour and N. Mahdou, Amalgamation of rings defined by Bezout-like conditions, J. Algebra Appl. 10 (2011), no. 6, 1343-1350. https://doi.org/10.1142/S0219498811005683
  20. I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464-491. https://doi.org/10.2307/1990591
  21. M. D. Larsen, W. J. Lewis, and T. S. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1974), 231-248. https://doi.org/10.2307/1997051
  22. N. Mahdou and M. A. S. Moutui, Amalgamated algebras along an ideal defined by Gaussian condition, J. Taibah Univ. for Science 9 (2015), 373-379. https://doi.org/10.1016/j.jtusci.2015.02.022
  23. M. Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York, 1962.
  24. J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, Inc., New York, 1979.