On $\alpha$-parallel short modules

Document Type : Original Paper

Authors

1 Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, Iran

2 ِDepartment of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Abstract

An $R$-module $M$ is called $\alpha$-parallel short modules, if for each parallel submodule $N$ to $M$ either $\pndim\, N \leq \alpha$ or $\ndim\, \frac{M}{N}\leq\alpha$ and $\alpha$ is the least ordinal

number with this property. Using this concept, we extend some of the basic results of $\alpha$-short modules

to $\alpha$-parallel short modules.

Also, we have studied the relationship between $\alpha$-parallel short modules and their parallel Noetherian dimension and we show that if $M$ is a $\alpha$-parallel short module, then $M$ has parallel Noetherian dimension and

$\alpha\leq\pndim\, M\leq \alpha+1$. Furthermore, we prove that if $M$ is an $\alpha$-parallel short

module with finite Goldie dimension, then $M$ has Noetherian dimension and $\alpha\leq\ndim\, M\leq\alpha+1$.

Keywords

Main Subjects

References

[1] Bilhan. G and Smith. P. F, Short modules and almost Noetherian modules, Math. Scand., 98 (2006),
12-18.
[2] Davoudian. M, Karamzadeh. O. A. S and Shirali. N, On α-short modules, Math. Scand., 114(1)
(2014), 26-37.
[3] Karamzadeh. O. A. S, Noetherian dimension, Ph.D. thesis, Exeter, 1974.
[4] Karamzadeh. O. A. S and Sajedinejad. A. R, Atomic modules, Comm. Algebra, 29(7) (2001), 2757-
2773.
[5] Karamzadeh. O. A. S and Shirali. N, On the countablity of Noetherian dimension of modules, Comm.
Algebra, 32(10) (2004), 4073-4083.
[6] Krause. G, On the Krull dimension of left Noetherian left Matlis rings, Math. Z., 118(3) (1970),
207-214.
[7] Lemonnier. B, Dimension de Krull et codeviation, Application au theorem dÉakin. Comm. Algebra,
6 (1978), 1647-1665.
[8] Rentschler. R and Gabriel. P, Sur la dimension des anneaux et ensembles ordonnés, CR Acad. Sci.
Paris, 265(2) (1967), 712-715.
[9] Shirali. M and Shirali. N, On parallel Krull dimension of modules, Comm. Algebra, 50(12) (2022),
5284-5295.
Journal of Advanced Mathematical Modeling
Volume 12, Issue 3
November 2022
Pages 437-447

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History

  • Receive Date: 26 June 2022
  • Revise Date: 31 October 2022
  • Accept Date: 01 November 2022

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