Document Type : Original Manuscript

Authors

Department of Mathematics, Faculty of Science, Universidad Nacional de Colombia - Sede Bogotá, Bogotá, D. C., Colombia.

Abstract

We present several results that establish the fusible and the regular left fusible properties of the family of noncommutative rings known as skew Poincar'e-Birkhoff-Witt extensions. Our treatment is based on the recent works of Ghashghaei and McGovern [13], and Kosan and Matczuk [31] concerning the left fusibleness and the regular left fusibleness of skew polynomial rings of automorphism type. Since the results formulated in this paper can be applied to algebraic structures more general than skew polynomial rings, our contribution to the theory of fusibleness is to cover more families of rings of interest in branches as quantum groups, noncommutative algebraic geometry and noncommutative differential geometry. We provide illustrative examples of the ideas developed here.

Keywords

1. E. Akalan, and L. Vas, Classes of almost clean rings, Algebr. Represent. Theory, 16(3) (2013), 843–857.
2. D. D. Anderson, and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra, 26(7) (1998), 2265–2275.
3. E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc., 18(4) (1974), 470–473.
4. V. A. Artamonov, Derivations of skew PBW extensions, Commun. Math. Stat., 3(4) (2015), 449–457. FUSIBLE PROPERTY OF SKEW PBW EXTENSIONS 27
5. A. Bell, and K. Goodearl, Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions, Pacific J. Math., 131(1) (1988), 13–37.
6. R. Berger, The Quantum Poincaré-Birkhoff-Witt Theorem, Comm. Math. Phys., 143(2) (1992), 215–234.
7. V. P. Camillo, and D. Khurana, A characterization of unit regular rings, Comm.
Algebra, 29(5) (2001), 2293–2295.
8. P. M. Cohn, Rings of zero-divisors, Proc. Amer. Math. Soc., 9 (1958), 914–919.
9. S. C. Coutinho, A Primer of Algebraic D-Modules, Cambridge University Press, 1995.
10. C. C. Faith, and P. Pillay, Classification of commutative FPF rings, Universidad de Murcia. Secretaria de Publicaciones e Intercambio Científico, Vol. 4, Notas de Matemática, 1990.
11. W. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Suárez, and H. Venegas, Skew PBW extensions. Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications, Springer Cham, 2020.
12. C. Gallego, and O. Lezama, Gröbner bases for ideals of -PBW extensions, Comm. Algebra, 39(1) (2011), 50–75.
13. E. Ghashghaei, and W. Wm, McGovern, Comm. Algebra, 45(3) (2017), 1151–
1165.
14. A. Giaquinto, and J. J. Zhang, Quantum Weyl Algebras, J. Algebra, 176(3) (1995), 861–881.
15. L. Gillman, and M. Jerison, Rings of continuous functions, Van Nostrand Company, Inc, 1 edition, 1960.
16. K. R. Goodearl, and R. B. Jr. Warfield, An Introduction to Noncommutative Noetherian Rings, Second Edition, London Mathematical Society Student Texts, 2004.
17. M. Hamidizadeh, E. Hashemi, and A. Reyes, A Classification of Ring Elements in Skew PBW extensions over compatible rings, Int. Electron. J. Algebra, 28(1)
(2020), 75–97.
18. E. Hashemi, Kh. Khalilnezhad, and A. Alhevaz, (; Δ)-compatible skew PBW extension ring, Kyungpook Math. J., 57(3) (2017), 401–417.
19. E. Hashemi, Kh. Khalilnezhad, and A. Alhevaz, Extensions of rings over 2- primal rings, Matematiche, 74(1) (2019), 141–162.
20. E. Hashemi, Kh. Khalilnezhad, and M. Ghadiri, Baer and quasi-Baer properties of skew PBW extensions, J. Algebr. Syst., 7(1) (2019), 1–24.
21. T. Hayashi, Q-analogues of Clifford and Weyl algebras-Spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys., 127(1)
(1990), 129–144.
22. J. Hernández, and A. Reyes, A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type, Ingeniería y Ciencia, 16(31) (2020), 27–52.
23. C. Y. Hong, N. K. Kim, and T. K. Kwak, On skew Armendariz rings, Comm. Algebra, 31(1) (2003), 103–122.
24. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra, 30(2) (2002), 751–761.
25. A. P. Isaev, P. N. Pyatov, and V. Rittenberg, Diffusion algebras, J. Phys. A.,
34(29) (2001), 5815–5834.
28 HIGUERA AND REYES
26. A. Jannussis, A. Leodaris, and R. Mignani, Non-Hermitian Realization of a Lie-deformed Heisenberg algebra, Physics Letters A, 197(3) (1995), 187–191.
27. D. Jordan, Iterated Skew Polynomial Rings and Quantum Groups, J. Algebra, 156(1) (1993), 194–218.
28. D. Jordan, Simple ambiskew polynomial rings, J. Algebra, 382, 46–70, 2013.
29. D. Jordan, The Graded Algebra Generated by Two Eulerian Derivatives, Algebr. Represent. Theory, 4(3) (2001), 249–275.
30. N. K. Kim, and Y. Lee, Armendariz rings and reduced rings, J. Algebra, 223(2) (2000), 477–488.
31. M. T. Koşan, and J. Matczuk, On fusible rings, Comm. Algebra, 47(9) (2019), 3789–3793.
32. T. K. Lee, and T. L. Wong, On Armendariz rings, Houston J. Math., 29(3) (2003), 583–593.
33. A. Leroy, and J. Matczuk, Goldie conditions for Ore extensions over semiprime rings, Algebr. Represent. Theory, 8(5) (2005), 679–688.
34. O. Lezama, Computation of point modules of finitely semi-graded rings, Comm. Algebra, 48(2) (2020), 862–878.
35. O. Lezama, J. P. Acosta, C. Chaparro, I. Ojeda, and C. Venegas, Ore and Goldie theorems for skew PBW extensions, Asian-Eur. J. Math., 6(04) (2013), 1350061.
36. O. Lezama, and C. Gallego, d-Hermite rings and skew PBW extensions, São Paulo J. Math. Sci., 10(1) (2016), 60–72.
37. O. Lezama, and J. Gómez, Koszulity and Point Modules of Finitely Semi-graded Rings and Algebras, Symmetry, 11(7) (2019), 881.
38. O. Lezama, and A. Reyes, Some Homological Properties of Skew PBW Extensions, Comm. Algebra, 42(3) (2014), 1200–1230.
39. M. Louzari, and A. Reyes, Minimal prime ideals of skew PBW extensions over
2-primal compatible rings, Rev. Colombiana Mat., 54(1) (2020), 34–63.
40. J. C. McConnell, and J. C. Robson, Noncommutative Noetherian rings, Second Edition, American Mathematical Society, 2001.
41. W. Wm. McGovern, Clean semiprime f-rings with bounded inversion, Comm. Algebra, 31(7) (2003), 3295–3304.
42. A. Niño, M. C. Ramírez, and A. Reyes, Associated prime ideals over skew PBW extensions, Comm. Algebra, 48(12) (2020), 5038–5055.
43. A. Niño, and A. Reyes, Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions, Algebra Discrete Math., 30(2) (2020), 207-
229.
44. A. Niño, and A. Reyes, Some ring theoretical properties for skew Poincaré- Birkhoff-Witt extensions, Bol. Mat., 24(2) (2017), 131–148.
45. O. Ore, Theory of Non-commutative Polynomials, Ann. of Math. (2), 34(3) (1933), 480–508.
46. M. B. Rege, and S. Chhawchharia, Armendariz Rings, Proc. Japan Acad. Ser. A Math. Sci., 73(1) (1997), 14–17.
47. A. Reyes, Armendariz modules over skew PBW extensions, Comm. Algebra, 47(3) (2019), 1248–1270.
48. A. Reyes, Uniform dimension over skew PBW extensions, Rev. Colombiana
Mat., 48(1) (2014), 79–96. FUSIBLE PROPERTY OF SKEW PBW EXTENSIONS 29
49. A. Reyes, and C. Rodríguez, The McCoy condition on skew PBW extensions, Commun. Math. Stat., 9(1) (2021), 1–21.
50. A. Reyes, and H. Suárez, Armendariz property for skew PBW extensions and their classical ring of quotients, Rev. Integr. Temas Mat., 34(2) (2016), 147–
168.
51. A. Reyes, and H. Suárez, Bases for quantum algebras and skew Poincaré-
Birkhoff-Witt extensions, Momento, 54(1) (2017), 54–75.
52. A. Reyes, and H. Suárez, Enveloping Algebra and Skew Calabi-Yau algebras
over Skew Poincaré-Birkhoff-Witt Extensions, Far East J. Math. Sci., 102(2)
(2017), 373–397.
53. A. Reyes, and H. Suárez, -PBW Extensions of Skew Armendariz Rings, Adv. Appl. Clifford Algebr., 27(4) (2017), 3197–3224.
54. A. Reyes, and H. Suárez, Skew Poincaré-Birkhoff-Witt extensions over weak zip rings, Beitr. Algebra Geom., 60(2) (2019), 197–216.
55. A. Rosenberg, Non-Commutative Algebraic Geometry and Representations of Quantized Weyl Algebras, Springer, 1995.
56. H. Yamane, A Poincaré-Birkhoff-Witt Theorem for Quantized Universal Enveloping Algebras of Type AN, Publ. RIMS Kyoto Univ., 25(3) (1989), 503–520.