ON α-SHORT TYPE MODULES

Document Type : Original Manuscript

Author

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

10.22044/jas.2024.13556.1754

Abstract

We introduce and study the concept of $\alpha $-short type modules.
Using this concept we extend some of the basic results of $\alpha $-short modules to $\alpha $-short type modules. We observe that if $M$ is an $\alpha $-short type module then the Noetherian dimension of $M$ is less than or equal to $\omega_{1}+\alpha+1

Keywords


 s
 
1. T. Albu and S. Rizvi, Chain conditions on Quotient finite dimensional modules, Comm. Algebra, 29(5) (2001), 1909–1928.
2. T. Albu and P. F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math., 29 (1999), 1153–1164.
3. T. Albu and P. Vamos, Global Krull dimension and global dual Krull dimension of valuation rings, In: Abelian Groups, Module Theory, and Topology, Marcel Dekker, (1998), 37–54.
4. G. Bilhan and P. F. Smith, Short modules and almost Noetherian modules, Math. Scand., 98 (2006), 12–18.
5. L. Chambless, N-Dimension and N-critical modules, application to Artinian modules, Comm. Algebra, 8(16) (1980), 1561–1592.
6. M. Davoudian, Dimension of uncountably generated submodules, Int. Electron. J. Algebra, 35 (2024), 149–159.
7. M. Davoudian and O. A. S. Karamzadeh, Artinian serial modules over commutative (or left Noetherian) rings are at most one step away from being Noetherian, Comm. Algebra, 44(9) (2016), 3907–3917.
8. M. Davoudian, O. A. S. Karamzadeh and N. Shirali, On α-short modules, Math. Scand., 114(1) (2014), 26–37.
9. R. Gordon, Gabriel and Krull dimension, In: Ring Theory (Proceeding of the Oklahoma Conference), Lecture Notes in Pure and Appl. Math., Vol. 7, Dekker, NewYork, (1974), 241–295.
10. R. Gordon and J. C. Robson, Krull dimension, Mem. Amer. Math. Soc., 1973.
11. J. Hashemi, O. A. S. Karamzadeh and N. Shirali, Rings over which the Krull dimension and Noetherian dimension of all modules coincide, Comm. Algebra, 37(2) (2009), 650– 662.
12. O. A. S. Karamzadeh, Noetherian-dimension, Ph.D. Thesis, Exeter, 1974.
13. O. A. S. Karamzadeh and M. Motamedi, On α-Dicc modules, Comm. Algebra, 22 (1994), 1933–1944.
14. O. A. S. Karamzadeh and A. R. Sajedinejad, Atomic modules, Comm. Algebra, 29(7) (2001), 2757–2773.
15. O. A. S. Karamzadeh and A. R. Sajedinejad, On the Loewy length and the Noetherian dimension of Artinian modules, Comm. Algebra, 30 (2002), 1077–1084.
16. O. A. S. Karamzadeh and N. Shirali, On the Countablity of Noetherian Dimension of Modules, Comm. Algebra, 32 (2004), 4073–4083.
17. D. Kirby, Dimension and length for Artinian modules, Quart. J. Math. Oxford., 41 (1990), 419–429.
18. B. Lemonnier, Deviation des ensembless et groupes totalement ordonnes, Bull. Sci. Math., 96 (1972), 289–303.
19. B. Lemonnier, Dimension de Krull et codeviation, Application au theorem d’Eakin, Comm. Algebra, 6 (1978), 1647–1665.
20. J. C. McConell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, New York, 1987.
21. R. N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford., 26 (1975), 269–273.