Publicado

2014-01-01

Uniform Dimension over Skew PBW Extensions

Dimensión uniforme de las extensiones PBW torcidas

DOI:

https://doi.org/10.15446/recolma.v48n1.45196

Palabras clave:

Non-commutative rings, Filtered and graded rings, PBW extensions, Uniform dimension, Nonsingular modules (en)
Anillos no conmutativos, anillos filtrados y graduados, extensiones PBW, dimensión uniforme, módulos no singulares (es)

Autores/as

  • Armando Reyes Universidad Nacional de Colombia
The aim of the present paper is to show that, under some conditions, the uniform dimension of a ring R is the same as the uniform dimension of a skew Poincaré-Birkhoff-Witt extension built on R.
El propósito de este artículo es mostrar que bajo ciertas condiciones, la dimensión uniforme de un anillo R coincide con la dimensión uniforme de una extensión Poincaré-Birkhoff-Witt torcida de R.

Uniform Dimension over Skew PBW Extensions

Dimensión uniforme de las extensiones PBW torcidas

ARMANDO REYES1

1Universidad Nacional de Colombia, Bogotá, Colombia. Email: mareyesv@unal.edu.co


Abstract

The aim of the present paper is to show that, under some conditions, the uniform dimension of a ring R is the same as the uniform dimension of a skew Poincaré-Birkhoff-Witt extension built on R.

Key words: Non-commutative rings, Filtered and graded rings, PBW extensions, Uniform dimension, Nonsingular modules.


2000 Mathematics Subject Classification: 16P40, 16P60, 16W70, 13N10, 16S36.

Resumen

El propósito de este artículo es mostrar que bajo ciertas condiciones, la dimensión uniforme de un anillo R coincide con la dimensión uniforme de una extensión Poincaré-Birkhoff-Witt torcida de R.

Palabras clave: Anillos no conmutativos, anillos filtrados y graduados, extensiones PBW, dimensión uniforme, módulos no singulares.


Texto completo disponible en PDF


References

[1] A. D. Bell and K. R. Goodearl, `Uniform Rank over Differential Operator Rings and Poincaré-Birkhoff-Witt extensions', Pacific Journal of Mathematics 131, 1 (1988), 13-37.

[2] C. Gallego and O. Lezama, `Gröbner Bases for Ideals of σ-PBW Extensions', Communications in Algebra 39, 1 (2011), 50-75.

[3] K. R. Goodearl, Nonsingular Rings and Modules, Pure and Applied Mathematics, New York, USA, 1976.

[4] K. R. Goodearl and T. Lenagan, `Krull Dimension of Differential Operator Rings III: Noncommutative Coeficients', Transactions of the American Mathematical Society 275, (1983), 833-859.

[5] P. Grzeszczuk, `Goldie Dimension of Differential Operator Rings', Communications in Algebra 16, 4 (1988), 689-701.

[6] T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, Graduate Texts in Mathematics 189, New York, USA, 1999.

[7] A. Leroy and J. Matczuk, `Goldie Conditions for Ore Extensions over Semiprime Rings', Algebras and Representation Theory 8, (2005), 679-688.

[8] O. Lezama and A. Reyes, `Some Homological Properties of Skew PBW Extensions', Communications in Algebra 42, (2014), 1200-1230.

[9] J. Matczuk, `Goldie Rank of Ore Extensions', Communications in Algebra 23, (1995), 1455-1471.

[10] J. McConnell and C. Robson, Non-commutative Noetherian Rings, with the Cooperation of L. W. Small., 2 edn, Graduate Studies in Mathematics. 30. American Mathematical Society (AMS), Providence, USA, 2001.

[11] V. A. Mushrub, `On the Goldie Dimension of Ore Extensions with Several Variables', Fundamentalnaya i Prikladnaya Matematika 7, (2001), 1107-1121.

[12] D. Quinn, `Embeddings of Differential Operator Rings and Goldie Dimension', Proceedings of the American Mathematical Society 102, 1 (1988), 9-16.

[13] A. Reyes, Ring and Module Theoretic Properties of σ-PBW Extensions, Ph.D. Thesis, Universidad Nacional de Colombia, 2013a.

[14] A. Reyes, `Gelfand-Kirillov Dimension of Skew PBW Extensions', Revista Colombiana de Matemáticas 47, 1 (2013b), 95-111.

[15] R. C. Shock, `Polynomial Rings over Finite-Dimensional Rings', Pacific Journal of Mathematics 42, (1972), 251-257.

[16] G. Sigurdsson, `Differential Operator Rings whose Prime Factors have Bounded Goldie Dimension', Archiv der Mathematik (Basel) 42, (1984), 348-353.


(Recibido en julio de 2013. Aceptado en enero de 2014)

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv48n1a05,
    AUTHOR  = {Reyes, Armando},
    TITLE   = {{Uniform Dimension over Skew \boldsymbol{PBW} Extensions}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2014},
    volume  = {48},
    number  = {1},
    pages   = {79--96}
}

Referencias

A. D. Bell and K. R. Goodearl, `Uniform Rank over Differential Operator Rings and Poincaré-Birkhoff-Witt extensions', Pacific Journal of Mathematics 131, 1 (1988), 13-37.

C. Gallego and O. Lezama, `Gröbner Bases for Ideals of σ-PBW Extensions', Communications in Algebra 39, 1 (2011), 50-75.

K. R. Goodearl, Nonsingular Rings and Modules, Pure and Applied Mathematics, New York, USA, 1976.

K. R. Goodearl and T. Lenagan, `Krull Dimension of Differential Operator Rings III: Noncommutative Coeficients', Transactions of the American Mathematical Society 275, (1983), 833-859.

P. Grzeszczuk, `Goldie Dimension of Differential Operator Rings', Communications in Algebra 16, 4 (1988), 689-701.

T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, Graduate Texts in Mathematics 189, New York, USA, 1999.

A. Leroy and J. Matczuk, `Goldie Conditions for Ore Extensions over Semiprime Rings', Algebras and Representation Theory 8, (2005), 679-688.

O. Lezama and A. Reyes, `Some Homological Properties of Skew PBW Extensions', Communications in Algebra 42, (2014), 1200-1230.

J. Matczuk, `Goldie Rank of Ore Extensions', Communications in Algebra 23, (1995), 1455-1471.

J. McConnell and C. Robson, Non-commutative Noetherian Rings, with the Cooperation of L. W. Small., 2 edn, Graduate Studies in Mathematics. 30. American Mathematical Society (AMS), Providence, USA, 2001.

V. A. Mushrub, `On the Goldie Dimension of Ore Extensions with Several Variables', Fundamentalnaya i Prikladnaya Matematika 7, (2001), 1107-1121.

D. Quinn, `Embeddings of Differential Operator Rings and Goldie Dimension', Proceedings of the American Mathematical Society 102, 1 (1988), 9-16.

A. Reyes, Ring and Module Theoretic Properties of σ-PBW Extensions, Ph.D. Thesis, Universidad Nacional de Colombia, 2013a.

A. Reyes, `Gelfand-Kirillov Dimension of Skew PBW Extensions', Revista Colombiana de Matemáticas 47, 1 (2013b), 95-111.

R. C. Shock, `Polynomial Rings over Finite-Dimensional Rings', Pacific Journal of Mathematics 42, (1972), 251-257.

G. Sigurdsson, `Differential Operator Rings whose Prime Factors have Bounded Goldie Dimension', Archiv der Mathematik (Basel) 42, (1984), 348-353.

Cómo citar

APA

Reyes, A. (2014). Uniform Dimension over Skew PBW Extensions. Revista Colombiana de Matemáticas, 48(1), 79–96. https://doi.org/10.15446/recolma.v48n1.45196

ACM

[1]
Reyes, A. 2014. Uniform Dimension over Skew PBW Extensions. Revista Colombiana de Matemáticas. 48, 1 (ene. 2014), 79–96. DOI:https://doi.org/10.15446/recolma.v48n1.45196.

ACS

(1)
Reyes, A. Uniform Dimension over Skew PBW Extensions. rev.colomb.mat 2014, 48, 79-96.

ABNT

REYES, A. Uniform Dimension over Skew PBW Extensions. Revista Colombiana de Matemáticas, [S. l.], v. 48, n. 1, p. 79–96, 2014. DOI: 10.15446/recolma.v48n1.45196. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/45196. Acesso em: 29 jun. 2022.

Chicago

Reyes, Armando. 2014. «Uniform Dimension over Skew PBW Extensions». Revista Colombiana De Matemáticas 48 (1):79-96. https://doi.org/10.15446/recolma.v48n1.45196.

Harvard

Reyes, A. (2014) «Uniform Dimension over Skew PBW Extensions», Revista Colombiana de Matemáticas, 48(1), pp. 79–96. doi: 10.15446/recolma.v48n1.45196.

IEEE

[1]
A. Reyes, «Uniform Dimension over Skew PBW Extensions», rev.colomb.mat, vol. 48, n.º 1, pp. 79–96, ene. 2014.

MLA

Reyes, A. «Uniform Dimension over Skew PBW Extensions». Revista Colombiana de Matemáticas, vol. 48, n.º 1, enero de 2014, pp. 79-96, doi:10.15446/recolma.v48n1.45196.

Turabian

Reyes, Armando. «Uniform Dimension over Skew PBW Extensions». Revista Colombiana de Matemáticas 48, no. 1 (enero 1, 2014): 79–96. Accedido junio 29, 2022. https://revistas.unal.edu.co/index.php/recolma/article/view/45196.

Vancouver

1.
Reyes A. Uniform Dimension over Skew PBW Extensions. rev.colomb.mat [Internet]. 1 de enero de 2014 [citado 29 de junio de 2022];48(1):79-96. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/45196

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