The McCoy Condition on Skew Poincaré–Birkhoff–Witt Extensions

  • Armando ReyesEmail author
  • Camilo Rodríguez


In this paper, we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincaré–Birkhoff–Witt extensions. As a consequence, we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.


McCoy ring Reversible ring Semicommutative ring Zip ring Skew Poincaré–Birkhoff–Witt extension 

Mathematics Subject Classification

16U80 16S36 16S38 



Research is supported by Grant HERMES CODE 30366, Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia, Sede Bogotá.


  1. 1.
    Annin, S.: Associated primes over skew polynomial rings. Commun. Algebra 30(5), 2511–2528 (2002)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Artamonov, A.: Derivations of skew PBW extensions. Commun. Math. Stat. 3(4), 449–457 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Artamonov, V., Lezama, O., Fajardo, W.: Extended modules and Ore extensions. Commun. Math. Stat. 4(2), 189–202 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Baser, M.: Ore extensions of zip and reversible rings. Commun. Fac. Sci. Univ. Ank. Ser. A1 55(1), 1–6 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Baser, M., Kwak, T.K., Lee, Y.: The McCoy condition on skew polynomial rings. Commun. Algebra 37(11), 4026–4037 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beachy, J.A., Blair, W.D.: Rings whose faithful left ideals are cofaithful. Pac. J. Math. 58(1), 1–13 (1975)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bell, H.E.: Near-rings in which each element is a power of itself. Bull. Aust. Math. Soc. 2(3), 363–368 (1970)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bell, A., Goodearl, K.: Uniform rank over differential operator rings and Poincaré–Birkhoff–Witt extensions. Pac. J. Math. 131(1), 13–37 (1988)zbMATHGoogle Scholar
  9. 9.
    Camillo, V., Nielsen, P.P.: McCoy rings and zero-divisors. J. Pure Appl. Algebra 212(3), 599–615 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cohn, P.M.: Reversible rings. Bull. Lond. Math. Soc. 31(6), 641–648 (1999)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cortes, W.: Skew polynomial extensions over zip rings. Int. J. Math. Math. Sci. 496720, (2008).
  12. 12.
    Curado, E.M.F., Hassouni, Y., Rego-Monteiro, M.A., Rodrigues, L.M.C.S.: Generalized Heisenberg algebra and algebraic method: the example of an infinite square-well potential. Phys. Lett. A. 372(19), 3350–3355 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Du, X.N.: On semicommutative rings and strongly regular rings. J. Math. Res. Expos. 14(1), 57–60 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Faith, C.: Rings with zero intersection property on annihilators: zip rings. Publ. Math. 33(2), 329–332 (1989)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Faith, C.: Annihilator ideals, associated primes and kasch-mccoy commutative rings. Commun. Algebra 19(7), 1867–1892 (1991)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ferran, C.: Zip rings and Malcev domains. Commun. Algebra 19(7), 1983–1991 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gallego, C., Lezama, O.: Gröbner bases for ideals of \(\sigma \)-PBW extensions. Commun. Algebra 39(1), 50–75 (2011)zbMATHGoogle Scholar
  18. 18.
    Gallego, C., Lezama, O.: Projective modules and Gröbner bases for skew PBW extensions. Diss. Math. 521, 1–50 (2017)zbMATHGoogle Scholar
  19. 19.
    Giaquinto, A., Zhang, J.J.: Quantum Weyl Algebras. J. Algebra 176(3), 861–881 (1995)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Habeb, J.M.: A note on zero commutative and duo rings. Math. J. Okayama Univ. 32(1), 73–76 (1990)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Habibi, M., Moussavi, A., Alhevaz, A.: The McCoy condition on Ore extensions. Commun. Algebra 41(1), 124–141 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hayashi, T.: \(Q\)-analogues of Clifford and Weyl algebras-Spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys. 127(1), 129–144 (1990)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hirano, Y.: On annihilator ideals of a polynomial ring over a noncommutative ring. J. Pure Appl. Algebra 168(1), 45–52 (2002)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hong, C.Y., Kim, N.K., Kwak, T.K.: Ore extensions of Baer and p.p.-rings. J. Pure Appl. Algebra 151(3), 215–226 (2000)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hong, C.Y., Kim, N.K., Kwak, T.K., Lee, Y.: Extensions of zip rings. J. Pure Appl. Algebra 195(3), 231–242 (2005)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Huh, C., Lee, Y., Smoktunowicz, A.: Armendariz rings and semicommutative rings. Comm. Algebra 30(2), 751–761 (2002)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Jannussis, A.: New Lie-deformed Heisenberg algebra. In: Lie-Lobachevski Colloquium on Lie Groups and Homogeneous Spaces, Tattu, pp. 26–30 Oct (1992)Google Scholar
  28. 28.
    Jannussis, A., Leodaris, A., Mignani, R.: Non-Hermitian realization of a Lie-deformed Heisenberg algebra. Phys. Lett. A. 197(3), 187–191 (1995)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Jategaonkar, A.V.: Localization in Noetherian rings, London Mathematical Society Lecture Note Series, 98. Cambridge University Press, Cambridge (1986)Google Scholar
  30. 30.
    Kim, N.K., Lee, Y.: Extensions of reversible rings. J. Pure Appl. Algebra 185(1–3), 207–223 (2003)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Krempa, J.: Some examples of reduced rings. Algebra Colloq. 3(4), 289–300 (1996)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Lezama, O., Acosta, J.P., Chaparro, C., Ojeda, I., Venegas, C.: Ore and Goldie theorems for skew PBW extensions. Asian Eur. J. Math. 6(4), 1350061-1–1350061-20 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lezama, O., Acosta, J.P., Reyes, A.: Prime ideals of skew PBW extensions. Rev. Un. Mat. Argent. 56(2), 39–55 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lezama, O., Latorre, E.: Non-commutative algebraic geometry of semi-graded rings. Int. J. Algebra Comput. 27(4), 361–389 (2017)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Lezama, O., Reyes, A.: Some homological properties of skew PBW extensions. Commun. Algebra 42(3), 1200–1230 (2014)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Marks, G.: A taxonomy of 2-primal rings. J. Algebra 266(2), 494–520 (2003)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Mason, G.: Reflexive ideals. Commun. Algebra 9(17), 1709–1724 (1981)MathSciNetzbMATHGoogle Scholar
  38. 38.
    McCoy, N.H.: Remarks on divisors of zero. Am. Math. Monthly 49(5), 286–295 (1942)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Nielsen, P.P.: Semi-commutativity and the McCoy condition. J. Algebra 298(1), 134–141 (2006)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Niño, D., Reyes, A.: Some ring theoretical properties of skew Poincaré–Birkhoff–Witt extensions. Bol. Mat. 24(2), 131–148 (2017)MathSciNetGoogle Scholar
  41. 41.
    Ore, O.: Theory of non-commutative polynomials. Ann. Math. Second Series 34(3), 480–508 (1933)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Reyes, A.: Skew PBW extensions of Baer, quasi-Baer, p.p. and p.q.-rings. Rev. Integr. Temas Mat. 33(2), 173–189 (2015)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Reyes, A.: Armendariz modules over skew pbw extensions. Commun. Algebra 47(3), 1248–1270 (2019)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Reyes, A., Suárez, H.: Some remarks about the cyclic homology of skew PBW extensions. Cienc. Desarro. 7(2), 99–107 (2016)Google Scholar
  45. 45.
    Reyes, A., Suárez, H.: A note on zip and reversible skew PBW extensions. Bol. Mat. (N.S.) 23(1), 71–79 (2016)MathSciNetGoogle Scholar
  46. 46.
    Reyes, A., Suárez, H.: Bases for quantum algebras and skew Poincaré–Birkhoff–Witt extensions. Momento 54(1), 54–75 (2017)Google Scholar
  47. 47.
    Reyes, A., Suárez, H.: PBW bases for some 3-dimensional skew polynomial algebras. Far East J. Math. Sci. 101(6), 1207–1228 (2017)zbMATHGoogle Scholar
  48. 48.
    Reyes, A., Suárez, H.: Enveloping algebra and skew Calabi–Yau algebras over skew Poincaré–Birkhoff–Witt extensions. Far East J. Math. Sci. 102(2), 373–397 (2017)zbMATHGoogle Scholar
  49. 49.
    Reyes, A., Suárez, H.: \(\sigma \)-PBW extensions of skew Armendariz rings. Adv. Appl. Clifford Algebr. 27(4), 3197–3224 (2017)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Reyes, A., Suárez, H.: A notion of compatibility for Armendariz and Baer properties over skew PBW extensions. Rev. Un. Mat. Argent. 59(1), 157–178 (2018)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Reyes, A., Suárez, H.: Skew Poincaré-Birkhoff-Witt extensions over weak zip rings. Beitr. Algebra Geom. 60(2), 197–216 (2019)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Reyes, A., Suárez, H.: Radicals and Köthe’s conjecture for skew PBW extensions. Commun. Math. Stat. (2019), (in press)Google Scholar
  53. 53.
    Reyes, A., Suárez, Y.: On the ACCP in skew Poincaré–Birkhoff–Witt extensions. Beitr Algebra Geom. 59(4), 625–643 (2018)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Suárez, H., Reyes, A.: A generalized Koszul property for skew PBW extensions. Far East J. Math. Sci. 101(2), 301–320 (2017)zbMATHGoogle Scholar
  55. 55.
    Suárez, H., Lezama, O., Reyes, A.: Calabi-Yau property for graded skew PBW extensions. Rev. Colomb. Mat. 51(2), 221–239 (2017)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Tuganbaev, A.A.: Rings close to regular mathematics and its applications, vol. 545. Kluwer Academic Publishers, New York (2002)zbMATHGoogle Scholar
  57. 57.
    Zelmanowitz, J.M.: The finite intersection property on annihilator right ideals. Proc. Am. Math. Soc. 57(2), 213–216 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Nacional de ColombiaSede BogotáColombia

Personalised recommendations