Research ArticleNo Access

Dimension on non-essential submodules

Maryam Davoudian

Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

https://doi.org/10.1142/S0219498819500890Cited by:3 (Source: Crossref)

Abstract

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an $R$ -module, where $R$ is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring $R$ with non-essential Noetherian dimension is uniform.

Communicated by A. Facchini

Keywords:

AMSC: 16P60, 16P20, 16P40

References

1. T. Albu and S. Rizvi, Chain conditions on quotient finite dimensional modules, Comm. Algebra 29(5) (2001) 1909–1928. Web of Science, Google Scholar
2. T. Albu and P. F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math. 29 (1999) 1153–1164. Web of Science, Google Scholar
3. T. Albu and P. F. Smith, Dual relative Krull dimension of modules over commutative rings, in Abelian Groups and Modules, eds. A. Facchini and C. Menini (Kluwer, 1995), pp. 1–15. Google Scholar
4. T. Albu and P. Vamos, Global Krull Dimension and Global Dual Krull Dimension of valuation Rings, in Abelian Groups, Modules Theory, and Topology (Marcel-Dekker, 1998), pp. 37–54. Google Scholar
5. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules (Springer-Verlag, 1992). Google Scholar
6. H. Bass, Descending chains and the Krull ordinal of commutative rings, J. Pure Appl. Algebra 1 (1971) 347–360. Google Scholar
7. L. Chambless, $N$ -Dimension and $N$ -critical modules, Application to Artinian modules, Comm. Algebra 8(16) (1980) 1561–1592. Web of Science, Google Scholar
8. M. Davoudian, Dimension of non-finitely generated submodules, Vietnam J. Math. 44 (2016) 817–827. Google Scholar
9. M. Davoudian, Modules satisfying double chain condition on non-finitely generated submodules have Krull dimension, Turkish J. Math. 41 (2017) 1570–1578. Web of Science, Google Scholar
10. M. Davoudian, Modules with chain condition on non-finitely generated submodules, Mediterr. J. Math. 15(1) (2018), 1–2. Web of Science, Google Scholar
11. M. Davoudian, A. Halali and N. Shirali, On $α$ -almost Artinian modules, Open Math. 14 (2016) 404–413. Web of Science, Google Scholar
12. 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 (2016) 3907–3917. Web of Science, Google Scholar
13. M. Davoudian, O. A. S. Karamzadeh and N. Shirali, On $α$ -short modules, Math. Scand. 114(1) (2014) 26–37. Google Scholar
14. A. Facchini, Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Mathematics, Vol. 167 (Brikhásser Verlag, Basel, 1998), reprinted in Modern Brikhäuser Classics (Brikhäuser Verlag, Basel, 2010). Google Scholar
15. C. Faith, Rings, Modules and Categories (Springer-Verlag, Berlin-New-York, 1975). Google Scholar
16. L. Fuchs, Torsion preradical and ascending Loewy series of modules, J. Reine Angew. Math. 239 (1970) 169–179. Web of Science, Google Scholar
17. R. Gordon and J. C. Robson, Krull Dimension, Memoirs of the American Mathematical Society, Vol. 133 (American Mathematical Society, 1973). Google Scholar
18. 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. Web of Science, Google Scholar
19. O. A. S. Karamzadeh, Noetherian-dimension, Ph.D. thesis, Exeter (1974). Google Scholar
20. O. A. S. Karamzadeh, When are Artinian modules countably generated? Bull. Iran. Math. Soc. 9 (1982) 171–176. Google Scholar
21. O. A. S. Karamzadeh and M. Motamedi, On $α$ - $D i c c$ modules, Comm. Algebra 22 (1994) 1933–1944. Web of Science, Google Scholar
22. O. A. S. Karamzadeh and M. Motamedi, A-Noetherian and Artinian modules, Comm. Algebra 23 (1995) 3685–3703. Web of Science, Google Scholar
23. O. A. S. Karamzadeh and Sh. Rahimpur, $λ$ -finitely embeded modules, Comm. Algebra 12(2) (2005) 281–292. Abstract, Google Scholar
24. O. A. S. Karamzadeh and A. R. Sajedinejad, Atomic modules, Comm. Algebra 29(7) (2001) 2757–2773. Web of Science, Google Scholar
25. 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. Web of Science, Google Scholar
26. O. A. S. Karamzadeh and N. Shirali, On the countablity of Noetherian dimension of modules, Comm. Algebra 32 (2004) 4073–4083. Web of Science, Google Scholar
27. D. Kirby, Dimension and length for Artinian modules, Quart. J. Math. Oxford. 41 (1990) 419–429. Web of Science, Google Scholar
28. G. Krause, Descending chains of submodules and the Krull dimension of Noetherian modules, J. Pure Appl. Algebra 3 (1973) 385–397. Google Scholar
29. B. Lemonnier, Deviation des ensembless et groupes totalement ordonnes, Bull. Sci. Math. 96 (1972) 289–303. Web of Science, Google Scholar
30. B. Lemonnier, Dimension de Krull et codeviation, Application au theorem d’Eakin, Comm. Algebra 6 (1978) 1647–1665. Web of Science, Google Scholar
31. J. C. McConell and J. C. Robson, Noncommutative Noetherian Rings (Wiley-Interscience, New York, 1987). Google Scholar
32. R. N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford 26 (1975) 269–273. Web of Science, Google Scholar
33. P. F. Smith and M. R. Vedadi, Modules with chain condition on non-essential submodules, Comm. Algebra 32(5) (2004) 1881–1894. Web of Science, Google Scholar