Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2024, Cilt: 35 Sayı: 35, 149 - 159, 09.01.2024
https://doi.org/10.24330/ieja.1385180

Öz

Kaynakça

  • T. Albu and S. T. Rizvi, Chain conditions on quotient finite dimensional modules, Comm. Algebra, 29(5) (2001), 1909-1928.
  • T. Albu and P. F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math., 29 (1999), 1153-1165.
  • T. Albu and P. Vamos, Global Krull dimension and global dual Krull dimension of valuation rings, Abelian Groups, Module Theory, and Topology (Padua, 1997), Lecture Notes in Pure and Appl. Math., vol. 201, Dekker, New York (1998), 37-54
  • L. Chambless, N-Dimension and N-critical modules. Application to Artinian modules, Comm. Algebra, 8(16) (1980), 1561-1592.
  • M. Davoudian, Dimension of non-finitely generated submodules, Vietnam J. Math., 44 (2016), 817-827.
  • M. Davoudian, Dimension on non-essential submodules, J. Algebra Appl., 18(5) (2019), 1950089 (11 pp).
  • M. Davoudian, On countably generated dimension, Algebra Colloq., 28 (2021), 361-366.
  • M. Davoudian, Modules with chain condition on uncountably generated submodules, J. Algebra Appl., 22(6) (2023), 2350134 (12 pp).
  • M. Davoudian, O. A. S. Karamzadeh and N. Shirali, On $\alpha $-short modules, Math. Scand., 114 (1) (2014), 26-37.
  • J. Dauns and L. Fuchs, Infinite Goldie dimensions, J. Algebra 115 (1988), 297-302.
  • C. Faith, Algebra. I. Rings, Modules and Categories, Springer-Verlag, Berlin-New York, 1981.
  • R. Gordon and J. C. Robson, Krull Dimension, Mem. Amer. Math. Soc., No. 133 American Mathematical Society, Providence, RI, 1973.
  • O. A. S. Karamzadeh, Noetherian Dimension, Ph.D. thesis, Exeter, 1974.
  • O. A. S. Karamzadeh and M. Motamedi, On $\alpha$-$DICC$ modules, Comm. Algebra, 22 (1994), 1933-1944.
  • O. A. S. Karamzadeh and A. R. Sajedinejad, Atomic modules, Comm. Algebra, 29(7) (2001), 2757-2773.
  • O. A. S. Karamzadeh and N. Shirali, On the countability of Noetherian dimension of modules, Comm. Algebra, 32 (2004), 4073-4083.
  • D. Kirby, Dimension and length for Artinian modules, Quart. J. Math. Oxford Ser. (2), 41 (1990), 419-429.
  • G. Krause, On fully left bounded left noetherian rings, J. Algebra, 23 (1972), 88-99.
  • B. Lemonnier, Deviation des ensembles et groupes abeliens totalement ordonnes, Bull. Sci. Math. (2), 96 (1972), 289-303.
  • B. Lemonnier, Dimension de Krull et codeviation, application au theorem d'Eakin, Comm. Algebra, 6 (1978), 1647-1665.
  • J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, Ltd., Chichester, 1987.
  • R. N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford Ser. (2), 26 (1975), 269-273.

Dimension of uncountably generated submodules

Yıl 2024, Cilt: 35 Sayı: 35, 149 - 159, 09.01.2024
https://doi.org/10.24330/ieja.1385180

Öz

In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated 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 and Noetherian dimension.
They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules.
It is proved that a quotient finite dimensional module $M$ has uncountably generated Krull dimension if and only if it has Krull dimension, but
the values of these dimensions might differ.
Similarly, a quotient finite dimensional module $M$ has uncountably generated Noetherian dimension if and only if it has Noetherian dimension.
We also show that the Noetherian dimension of a quotient finite dimensional module $M$ with uncountably generated Noetherian dimension $\beta$ is less than or equal to $\omega _{1}+\beta $, where $\omega_{1}$ is the first uncountable ordinal number.

Kaynakça

  • T. Albu and S. T. Rizvi, Chain conditions on quotient finite dimensional modules, Comm. Algebra, 29(5) (2001), 1909-1928.
  • T. Albu and P. F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math., 29 (1999), 1153-1165.
  • T. Albu and P. Vamos, Global Krull dimension and global dual Krull dimension of valuation rings, Abelian Groups, Module Theory, and Topology (Padua, 1997), Lecture Notes in Pure and Appl. Math., vol. 201, Dekker, New York (1998), 37-54
  • L. Chambless, N-Dimension and N-critical modules. Application to Artinian modules, Comm. Algebra, 8(16) (1980), 1561-1592.
  • M. Davoudian, Dimension of non-finitely generated submodules, Vietnam J. Math., 44 (2016), 817-827.
  • M. Davoudian, Dimension on non-essential submodules, J. Algebra Appl., 18(5) (2019), 1950089 (11 pp).
  • M. Davoudian, On countably generated dimension, Algebra Colloq., 28 (2021), 361-366.
  • M. Davoudian, Modules with chain condition on uncountably generated submodules, J. Algebra Appl., 22(6) (2023), 2350134 (12 pp).
  • M. Davoudian, O. A. S. Karamzadeh and N. Shirali, On $\alpha $-short modules, Math. Scand., 114 (1) (2014), 26-37.
  • J. Dauns and L. Fuchs, Infinite Goldie dimensions, J. Algebra 115 (1988), 297-302.
  • C. Faith, Algebra. I. Rings, Modules and Categories, Springer-Verlag, Berlin-New York, 1981.
  • R. Gordon and J. C. Robson, Krull Dimension, Mem. Amer. Math. Soc., No. 133 American Mathematical Society, Providence, RI, 1973.
  • O. A. S. Karamzadeh, Noetherian Dimension, Ph.D. thesis, Exeter, 1974.
  • O. A. S. Karamzadeh and M. Motamedi, On $\alpha$-$DICC$ modules, Comm. Algebra, 22 (1994), 1933-1944.
  • O. A. S. Karamzadeh and A. R. Sajedinejad, Atomic modules, Comm. Algebra, 29(7) (2001), 2757-2773.
  • O. A. S. Karamzadeh and N. Shirali, On the countability of Noetherian dimension of modules, Comm. Algebra, 32 (2004), 4073-4083.
  • D. Kirby, Dimension and length for Artinian modules, Quart. J. Math. Oxford Ser. (2), 41 (1990), 419-429.
  • G. Krause, On fully left bounded left noetherian rings, J. Algebra, 23 (1972), 88-99.
  • B. Lemonnier, Deviation des ensembles et groupes abeliens totalement ordonnes, Bull. Sci. Math. (2), 96 (1972), 289-303.
  • B. Lemonnier, Dimension de Krull et codeviation, application au theorem d'Eakin, Comm. Algebra, 6 (1978), 1647-1665.
  • J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, Ltd., Chichester, 1987.
  • R. N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford Ser. (2), 26 (1975), 269-273.
Toplam 22 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebir ve Sayı Teorisi
Bölüm Makaleler
Yazarlar

Maryam Davoudıan

Erken Görünüm Tarihi 10 Kasım 2023
Yayımlanma Tarihi 9 Ocak 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 35 Sayı: 35

Kaynak Göster

APA Davoudıan, M. (2024). Dimension of uncountably generated submodules. International Electronic Journal of Algebra, 35(35), 149-159. https://doi.org/10.24330/ieja.1385180
AMA Davoudıan M. Dimension of uncountably generated submodules. IEJA. Ocak 2024;35(35):149-159. doi:10.24330/ieja.1385180
Chicago Davoudıan, Maryam. “Dimension of Uncountably Generated Submodules”. International Electronic Journal of Algebra 35, sy. 35 (Ocak 2024): 149-59. https://doi.org/10.24330/ieja.1385180.
EndNote Davoudıan M (01 Ocak 2024) Dimension of uncountably generated submodules. International Electronic Journal of Algebra 35 35 149–159.
IEEE M. Davoudıan, “Dimension of uncountably generated submodules”, IEJA, c. 35, sy. 35, ss. 149–159, 2024, doi: 10.24330/ieja.1385180.
ISNAD Davoudıan, Maryam. “Dimension of Uncountably Generated Submodules”. International Electronic Journal of Algebra 35/35 (Ocak 2024), 149-159. https://doi.org/10.24330/ieja.1385180.
JAMA Davoudıan M. Dimension of uncountably generated submodules. IEJA. 2024;35:149–159.
MLA Davoudıan, Maryam. “Dimension of Uncountably Generated Submodules”. International Electronic Journal of Algebra, c. 35, sy. 35, 2024, ss. 149-5, doi:10.24330/ieja.1385180.
Vancouver Davoudıan M. Dimension of uncountably generated submodules. IEJA. 2024;35(35):149-5.