Modules satisfying double chain condition on uncountably generated submodules

Document Type : Original Paper

Author

Department of mathematics, Shahid Chamran university of Ahvaz, Ahvaz, Iran

Abstract

In this article, we study modules that satisfy the double infinite chain condition on uncountably generated submodules, briefly called $u.c.g.-DICC$ modules. We show that if a quotient finite dimensional module $M$ satisfies the double infinite chain condition on uncountably generated submodules, then it has Krull dimension. We study submodules $N$ of a module $M$ such that whenever $\frac{M}{N}$ satisfies the double infinite chain condition so does $M$.
Moreover, we observe that an $\alpha $-atomic module, where $\alpha>\omega_{1}$ is an ordinal number, satisfies the previous chain if and only if it satisfies the descending chain condition on uncountably generated submodules.

Highlights

[1] Albu, T. and Vamos, P., 1998. Global Krull dimension and Global dual Krull dimension of
valuation rings, abelian groups, modules theory, and topology. Proc Marcel-Dekker, pp. 37-54.
https://doi.org/10.1201/9780429187605.
[2] Anderson, F. W. and Fuller, K. R., 1973. Rings and Categories of Modules. New York, NY, USA:
Springer-Verlag.
[3] Chambless, L., 1980. N-dimension and N-critical modules, application to Artinian modules. Comm Algebra, 8, pp. 1561-1592. https://doi.org/10.1080/00927878008822534.
[4] Contessa, M., 1987. On modules with DICC. J Algebra, 107, pp. 75-81. https://doi.org/10.1016/0021-8693(87)90074-3

[5] Contessa, M., 1987. On DICC rings. J Algebra, 105, pp. 429-436. https://doi.org/10.1016/0021-
8693(87)90206-7.
[6] Contessa, M., 1986. On rings and modules with DICC. J Algebra 101, pp. 489-496.
https://doi.org/10.1016/0021-8693(86)90207-3.
[7] Davoudian, M., 2018. Chain condition on non-finitelg generated submodules. Mediterr. J. Math., 15 (1), pp. 1-12. https://doi.org/10.1007/s00009-017-1047-y.
[8] Davoudian, M., Karamzadeh, O. A. S. and Shirali, N., 2014. On α-short modules. Math Scand, 114, pp. 26-37. http://dx.doi.org/10.7146/math.scand.a-16638.
[9] Davoudian, M., 2023. Chain condition on uncountably generated submodules. Journal of Algebra and its Applicatins, 22(6), 2350134, pp. 1-12. https://doi.org/10.1142/S0219498823501347.
[10] Davoudian, M., 2017. Modules satisfying double chain condition on nonfinitely generated submodules have Krull dimension. Turk J Math 41, pp. 1570-1578. https://doi:10.3906/mat-1501-14.
[11] Douns, J. and Fuchs, L., 1988. Infinite Goldie dimension. J. Algebra, 115, pp. 297-302.
https://doi.org/10.1016/0021-8693(88)90257-8.
[12] Gordon, R. and Robson, J. C., 1973. Krull dimension. Mem Amer Math Soc, Series 133.
[13] Karamzadeh, O. A. S. and Motamedi, M., 1994. On α-Dicc modules. Comm Algebra, 22, pp. 1933- 1944. https://doi.org/10.1080/00927879408824948.
[14] Karamzadeh, O. A. S. and Sajedinejad, A. R., 2001. Atomic modules. Comm Algebra, 29, pp. 2757-2773. https://doi.org/10.1081/AGB-4985.
[15] Karamzadeh, O. A. S. and Shirali, N., 2004. On the countability of Noetherian dimension of modules. Comm Algebra, 32, pp. 4073-4083. https://doi.org/10.1081/AGB-200028238.
[16] Lemonnier, B., 1972. Deviation des ensembless et groupes totalement ordonnes. Bull Sci Math 96, pp. 289-303.
[17] Lemonnier, B., 1987. Dimension de Krull et codeviation, Application au theorem d’Eakin. Comm Algebra, 6, pp. 1647-1665. https://doi.org/10.1080/00927877808822313
[18] Osofsky, B. L., 1987. Double Infinite Chain Conditions. In: Gobel R, Walker EA, editors. Abelian
Group Theory, New York, NY, USA: Gordon and Breach Science Publishers, pp. 451-456.
[19] Rahimpour, Sh., 2002. Double infinite chain condition on small and f.g. submodules. Far East J
Math Sci 6, pp. 167-177

 

Keywords

Main Subjects

References

[1] Albu, T. and Vamos, P., 1998. Global Krull dimension and Global dual Krull dimension of
valuation rings, abelian groups, modules theory, and topology. Proc Marcel-Dekker, pp. 37-54.
https://doi.org/10.1201/9780429187605.
[2] Anderson, F. W. and Fuller, K. R., 1973. Rings and Categories of Modules. New York, NY, USA:
Springer-Verlag.
[3] Chambless, L., 1980. N-dimension and N-critical modules, application to Artinian modules. Comm Algebra, 8, pp. 1561-1592. https://doi.org/10.1080/00927878008822534.
[4] Contessa, M., 1987. On modules with DICC. J Algebra, 107, pp. 75-81. https://doi.org/10.1016/0021-8693(87)90074-3
[5] Contessa, M., 1987. On DICC rings. J Algebra, 105, pp. 429-436. https://doi.org/10.1016/0021-
8693(87)90206-7.
[6] Contessa, M., 1986. On rings and modules with DICC. J Algebra 101, pp. 489-496.
https://doi.org/10.1016/0021-8693(86)90207-3.
[7] Davoudian, M., 2018. Chain condition on non-finitelg generated submodules. Mediterr. J. Math., 15 (1), pp. 1-12. https://doi.org/10.1007/s00009-017-1047-y.
[8] Davoudian, M., Karamzadeh, O. A. S. and Shirali, N., 2014. On α-short modules. Math Scand, 114, pp. 26-37. http://dx.doi.org/10.7146/math.scand.a-16638.
[9] Davoudian, M., 2023. Chain condition on uncountably generated submodules. Journal of Algebra and its Applicatins, 22(6), 2350134, pp. 1-12. https://doi.org/10.1142/S0219498823501347.
[10] Davoudian, M., 2017. Modules satisfying double chain condition on nonfinitely generated submodules have Krull dimension. Turk J Math 41, pp. 1570-1578. https://doi:10.3906/mat-1501-14.
[11] Douns, J. and Fuchs, L., 1988. Infinite Goldie dimension. J. Algebra, 115, pp. 297-302.
https://doi.org/10.1016/0021-8693(88)90257-8.
[12] Gordon, R. and Robson, J. C., 1973. Krull dimension. Mem Amer Math Soc, Series 133.
[13] Karamzadeh, O. A. S. and Motamedi, M., 1994. On α-Dicc modules. Comm Algebra, 22, pp. 1933- 1944. https://doi.org/10.1080/00927879408824948.
[14] Karamzadeh, O. A. S. and Sajedinejad, A. R., 2001. Atomic modules. Comm Algebra, 29, pp. 2757-2773. https://doi.org/10.1081/AGB-4985.
[15] Karamzadeh, O. A. S. and Shirali, N., 2004. On the countability of Noetherian dimension of modules. Comm Algebra, 32, pp. 4073-4083. https://doi.org/10.1081/AGB-200028238.
[16] Lemonnier, B., 1972. Deviation des ensembless et groupes totalement ordonnes. Bull Sci Math 96, pp. 289-303.
[17] Lemonnier, B., 1987. Dimension de Krull et codeviation, Application au theorem d’Eakin. Comm Algebra, 6, pp. 1647-1665. https://doi.org/10.1080/00927877808822313
[18] Osofsky, B. L., 1987. Double Infinite Chain Conditions. In: Gobel R, Walker EA, editors. Abelian
Group Theory, New York, NY, USA: Gordon and Breach Science Publishers, pp. 451-456.
[19] Rahimpour, Sh., 2002. Double infinite chain condition on small and f.g. submodules. Far East J
Math Sci 6, pp. 167-177
 

Files

History

  • Receive Date: 19 November 2023
  • Revise Date: 01 February 2024
  • Accept Date: 02 May 2024

Share

How to cite

Statistics