Ideales primos minimales de extensiones PBW torcidas sobre anillos compatibles 2-primal

• Louzari, Mohamed ^[1] ; Reyes, Armando ^[2]
  1. [1] Abdelmalek Essaâdi University

     Abdelmalek Essaâdi University

     Marruecos

     
  2. [2] Universidad Nacional de Colombia

     Universidad Nacional de Colombia

     Colombia

     
• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 54, Nº. 1, 2020, págs. 39-63
• Idioma: español
• DOI: 10.15446/recolma.v54n1.89788
• Títulos paralelos:
  • Minimal prime ideals of skew PBW extensions over 2-primal compatible rings
• Enlaces
  • Texto completo
• Resumen
  • español

    En este artículo, caracterizamos las unidades de las extensiones PBW torcidas sobre anillos compatibles. Con este propósito, recordamos la transferencia de la propiedad 2-primal para estas extensiones. Como una consecuencia de nuestro tratamiento, los resultados establecidos aquí generalizan aquellos correspondientes para anillos conmutativos y extensiones de Ore de tipo inyectivo. De esta manera, presentamos nuevos resultados para anillos no conmutativos de tipo polinomial no considerados antes en la literatura.

  • English

    In this paper, we characterize the units of skew PBW extensions over compatible rings. With this aim, we recall the transfer of the property of being 2-primal for these extensions. As a consequence of our treatment, the results established here generalize those corresponding for commutative rings and Ore extensions of injective type. In this way, we present new results for several noncommutative rings of polynomial type not considered before in the literature.

• Referencias bibliográficas
  • S. Annin, Associated and Attached Primes Over Noncommutative Rings, PhD thesis, University of California, Berkeley, 2002.
  • S. Annin, Associated primes over Ore extension rings, J. Algebra Appl. 3 (2004), no. 2, 193-205.
  • V. A. Artamonov, Derivations of skew PBW extensions, Commun. Math. Stat. 3 (2015), no. 4, 449-457.
  • V. A. Artamonov, O. Lezama, and W. Fajardo, Extended modules and Ore extensions, Commun. Math. Stat 4 (2016), no. 2, 189-202.
  • V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4 (1992), no. 1, 75-97.
  • A. Bell and K. Goodearl, Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions, Pacific J. Math. 131 (1988),...
  • J. Bueso, J. Gómez-Torrecillas, and A. Verschoren, Algorithmic Methods in Non-commutative algebra: Applications to Quantum Groups, Kluwer...
  • M. Ferrero and K. Kishimoto, On differential rings and skew polynomials, Comm. Algebra 13 (1985), no. 2, 285-304.
  • C. Gallego and O. Lezama, Gröbner bases for ideals of o-PBW extensions, Comm. Algebra 39 (2011), no. 1, 50-75.
  • E. Hashemi, Compatible ideals and radicals of Ore extensions, New York J. Math. 12 (2006), 349-356.
  • E. Hashemi, K. Khalilnezhad, and A. Alhevaz, (o; d)-Compatible Skew PBW Extension Ring, Kyungpook Math. J. 57 (2017), no. 3, 401-417.
  • E. Hashemi, K. Khalilnezhad, and A. Alhevaz, Extensions of rings over 2-primal rings, Le Matematiche LXXIV (2019), no. I, 141-162.
  • E. Hashemi, K. Khalilnezhad, and M. Ghadiri Herati, Baer and quasi-Baer properties of skew PBW extensions, J. Algebraic Systems 7 (2019),...
  • E. Hashemi and A. Moussavi, Polinomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (2005), no. 3, 207-224.
  • E. Hashemi, A. Moussavi, and H. Haj Seyyed Javadi, Polinomial Ore extensions of Baer and p.p.-rings, Bull. Iranian Math. Soc. 29 (2003), no....
  • C. Y. Hong, J. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000), no. 3, 215-226.
  • C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Rigid ideals and radicals of Ore extensions, Algebra Colloq. 12 (2005), no. 3, 399-412.
  • J. Jaramillo and A. Reyes, Symmetry and Reversibility Properties for Quantum Algebras and Skew Poincaré-Birkhoff-Witt Extensions, Ingeniería...
  • H. Jiménez and O. Lezama, Gröbner bases of modules over o-PBW extensions, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 32 (2016), no. 1, 39-66.
  • N. K. Kim and T. K. Kwak, Minimal prime ideals in 2-primal rings, Math. Japon 50 (1999), no. 3, 415-420.
  • J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.
  • T. Y. Lam, A. Leroy, and J. Matczuk, Primeness, Semiprimeness and Prime Radical of Ore extensions, Comm. Algebra 25 (1997), no. 8, 2459-2506.
  • O. Lezama, J. P. Acosta, and A. Reyes, Prime ideals of skew PBW extensions, Rev. Un. Mat. Argentina 56 (2015), no. 2, 39-55.
  • O. Lezama and C. Gallego, d-Hermite rings and skew PBW extensions, Sao Paulo J. Math. Sci. 10 (2016), no. 1, 60-72.
  • O. Lezama and C. Gallego, Projective modules and Gröbner bases for skew PBW extensions, Dissertationes Math. 521 (2017), 1-50.
  • O. Lezama and A. Reyes, Some Homological Properties of Skew PBW Extensions, Comm. Algebra 42 (2014), no. 3, 1200-1230.
  • O. Lezama and H. Venegas, Some homological properties of skew PBW extensions arising in non-commutative algebraic geometry, Discuss. Math....
  • G. Marks, Skew polynomial rings over 2-primal rings, Comm. Algebra 27 (1999), no. 9, 4411-4423.
  • G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123.
  • G. Marks, A taxonomy of 2-primal rings, J. Algebra 266 (2003), no. 2, 494-520.
  • A. R. Nasr-Isfahani, Ore extensions of 2-primal rings, J. Algebra Appl. 13 (2014), no. 3, 1350117-1 - 1350117-6.
  • A. Niño and A. Reyes, Some ring theoretical properties of skew Poincaré-Birkhoff-Witt extensions, Bol. Mat. 24 (2017), no. 2, 141-148.
  • A. Niño and A. Reyes, Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions, Algebra Discrete Math. (2020), To...
  • O. Ore, Theory of Non-Commutative Polynomials, Ann. of Math. Second Series 34 (1933), no. 3, 480-508.
  • A. Reyes, Uniform Dimension over Skew PBW extensions, Rev. Colombiana Mat. 48 (2014), no. 1, 79-96.
  • A. Reyes, Skew PBW extensions of Baer, quasi-Baer, p.p. and p.q.-rings, Rev. Integr. Temas Mat. 33 (2015), no. 2, 173-189.
  • A. Reyes, Armendariz modules over skew PBW extensions, Comm. Algebra 47 (2019), no. 3, 1248-1270.
  • A. Reyes and C. Rodríguez, The McCoy Condition on Skew Poincaré-Birkhoff-Witt Extensions, Commun. Math. Stat. (2019), https://doi.org/10.1007/s40304-019-00184-5.
  • A. Reyes and H. Suárez, A note on zip and reversible skew PBW extensions, Bol. Mat. 23 (2016), no. 1, 71-79.
  • A. Reyes and H. Suárez, Bases for quantum algebras and skew Poincaré-Birkhoff-Witt extensions, Momento 54 (2017), 54-75.
  • A. Reyes and H. Suárez, PBW bases for some 3-dimensional skew polynomial algebras, Far East J. Math. Sci. (FJMS) 101 (2017), no. 6, 1207-1228.
  • A. Reyes and H. Suárez, o-PBW Extensions of Skew Armendariz Rings, Adv. Appl. Clifford Algebr. 27 (2017), no. 4, 3197-3224.
  • A. Reyes and H. Suárez, A notion of compatibility for Armendariz and Baer properties over skew PBW extensions, Rev. Un. Mat. Argentina 59...
  • A. Reyes and H. Suárez, Radicals and Köthe's conjecture for skew PBW extensions, Commun. Math. Stat. (2019), https://doi.org/10.1007/s40304-019-00189-0.
  • A. Reyes and H. Suárez, Skew Poincaré-Birkhoff-Witt extensions over weak zip rings, Beitr. Algebra Geom. 60 (2019), no. 2, 197-216.
  • A. Reyes and H. Suárez, Skew Poincaré-Birkhoff-Witt extensions over weak compatible rings, J. Algebra Appl. (2020), https://doi.org/10.1142/S0219498820502254.
  • A. Reyes and Y. Suárez, On the ACCP in skew Poincaré-Birkhoff-Witt extensions, Beitr. Algebra Geom. 59 (2018), no. 4, 625-643.
  • A. L. Rosenberg, Non-commutative Algebraic Geometry and Representations of Quantized Algebras, 330 ed., Kluwer Academic Publishers, 1995.
  • G. Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973-74), 43-60.
  • A. Smoktunowicz, Polynomial rings over nil rings need not be nil, J. Algebra 223 (2000), no. 2, 427-436.
  • H. Suárez, O. Lezama, and A. Reyes, Calabi-Yau property for graded skew PBW extensions, Rev. Colombiana Mat. 51 (2017), no. 2, 221-239.
  • H. Suárez and A. Reyes, Nakayama Automorphism of Some Skew PBW Extensions, Ingeniería y Ciencia 15 (2019), no. 29, 157-177.
  • A. B. Tumwesigye, J. Richter, and S. Silvestrov, Centralizers in PBW extensions, (2019), https://arxiv.org/pdf/1910.11177.pdf.
  • Y. Wang and W. Chen, Minimal Prime Ideals and Units in 2-Primal Ore Extensions, J. Math. Res. Appl. 38 (2018), no. 4, 377-383.