Uniform Dimension over Skew PBW Extensions

Dimensión uniforme de las extensiones PBW torcidas

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.

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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

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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.

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

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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.

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Texto completo disponible en PDF

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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.

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

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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}
}

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