Modules satisfying double chain condition on non-small submodules

Document Type : Original Paper

Authors

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 non-small submodules, denoted by ns-DICC. Using this concept we extend some of the basic results of DICC modules to ns-DICC modules. We show that if an R-module M satisfies the double infinite chain condition on non-small submodules, then M has non-small Krull dimension. Moreover, we observe that an R-module M is ns-DICC if and only if for any non-small summand A of M, either A satisfies the descending chain condition on non-small submodules, or M/A is Noetherian.

Keywords

Main Subjects

References

[1] T. Albu, S. Rizvi,Chain conditions on quotient finite dimensional modules, Comm. Algebra. 29(5) (2001) 1909–1928.
[2] T. Albu, P. F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math. 29 (1999) 1153–1164.
[3] I. Al. khazzi, P. F. Smith, Modules with chain condition on supperfluous submodules, Commun. Algebra.19 (8) (1991) 2332-2351.
[4] M. Contessa, On modules with DICC, J Algebra. 107 (1987) 75–81.
[5] M. Contessa, On DICC rings, J Algebra. 105 (1987) 429–436.
[6] M. Davoudian, Modules with chain condition on non-finitely generated submodules, Mediterr. J. Math.15(1)(2018). https://doi.org/10.1007/s00009-017-1047-y
[7] M. Davoudian, Dimension of non-finitely generated submodules, Vietnam J. Math. 44 (2016) 817–827.[8] M. Davoudian, Dimension on non-small submodules, Bull. Iran. Math. Soc. 45 (2019) 1758–1793.[9] M. Davoudian, Dimension on non-essential submodules, J. Algebra Appl. 18(5) (2019).https://doi.org/10.1142/S0219498819500890
[10] M. Davoudian, Modules satisfying double chain condition on non-finitely generated submoduleshave Krull dimension, Turk. J. Math. 41 (2017) 1570–1578.
[11] R. Gordon, J. C. Robson, Krull dimension, Mem. Amer. Math. Soc. 133, 1973.
[12] O. A. S. Karamzadeh, Noetherian-dimension, Ph.D. thesis, Exeter, 1974.
[13] O. A. S. Karamzadeh, M. Motamedi, On α-Dicc modules, Comm. Algebra. 22 (1994) 1933–1944.[14] G. Krause, Ascending chains of submodules and the Krull dimension of Noetherian modules, J. PureAppl. Algebra. 3 (1973) 385–397.
[15] B. Lemonnier, Deviation des ensembless et groupes totalement ordonnes, Bull. Sci. Math. 96 (1972)289–303.[16] C. Lomp, On dual Goldie dimension, Ph. D. thesis, Dussldorf, 1996.
[17] P. F. Smith, M. R. Vedadi, Modules with chain condition on non-essential submodules, Comm.
Algebra. 32(5) (2004) 1881–1894.
Journal of Advanced Mathematical Modeling
Volume 12, Issue 3
November 2022
Pages 316-322

Files

History

  • Receive Date: 12 March 2021
  • Revise Date: 06 May 2022
  • Accept Date: 20 August 2022

Share

How to cite

Statistics