On weak annihilators and nilpotent associated primes of skew PBW extensions

References

• Annin, S. (2004). Associated primes over Ore extension rings. J. Algebra Appl. 3(2):193–205. DOI: 10.1142/S0219498804000782.
   Web of Science ®Google Scholar
• Annin, S. (2008). Attached primes over noncommutative rings. J. Pure Appl. Algebra 212(3):510–521. DOI: 10.1016/j.jpaa.2007.05.024.
   Web of Science ®Google Scholar
• Annin, S. (2011). Attached primes under skew polynomial extensions. J. Algebra Appl. 10(3):537–547. DOI: 10.1142/S0219498811004689.
   Web of Science ®Google Scholar
• Armendariz, E. P. (1974). A note on extensions of Baer and p.p.-rings. J. Austral. Math. Soc. 18:470–473. DOI: 10.1017/S1446788700029190.
   Google Scholar
• Artamonov, V. A. (2015). Derivations of skew PBW extensions. Commun. Math. Stat. 3(4):449–457. DOI: 10.1007/s40304-015-0067-9.
   Web of Science ®Google Scholar
• Bavula, V. V. (1992). Generalized Weyl algebras and their representations. St. Petersburg Math. J. 4(1):71–92.
   Google Scholar
• Bavula, V. V. (1996). Global dimension of Generalized Weyl algebras. Can. Math. Soc. Conf. Proc. 18:81–107.
   Google Scholar
• Bavula, V. V. (2021). Description of bi-quadratic algebras on 3 generators with PBW basis. https://arxiv.org/abs/2112.11105v1
   Google Scholar
• Beachy, J. A., Blair, W. D. (1975). Rings whose faithful left ideals are cofaithful. Pacific J. Math. 58(1):1–13. DOI: 10.2140/pjm.1975.58.1.
   Web of Science ®Google Scholar
• Bell, A. D., Goodearl, K. R. (1988). Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions. Pacific J. Math. 131(1):13–37. DOI: 10.2140/pjm.1988.131.13.
   Web of Science ®Google Scholar
• Bell, A. D., Smith, S. P. (1990). Some 3-dimensional skew polynomial rings. University of Wisconsin, Milwaukee, preprint.
   Google Scholar
• Benkart, G. (1998). Down-up algebras and Witten’s deformation of the universal enveloping algebra of sl2 . In: Geun Hahn, S., Chul Myung, H., Zelmanov, E., eds. Recent Progress in Algebra, Contemporary Mathematics, Vol. 224. Providence: American Mathematical Society, pp. 29–45.
   Google Scholar
• Benkart, G., Roby, T. (1998). Down-up algebras. J. Algebra 209(1):305–344. DOI: 10.1006/jabr.1998.7511.
   Web of Science ®Google Scholar
• Bhat, V. K. (2010). Associated prime ideals of weak σ-rigid rings and their extensions. Algebra Discrete Math. 10(1):8–17.
   Google Scholar
• Birkenmeier, G. F., Heatherly, H. E., Lee, E. K. (1993). Completely prime ideals and associated radicals. In: Jain, S. K., Rizvi, S. T., eds. Ring Theory. Granville, OH, 1992. Singapore and River Edge: World Scientific, pp. 102–129.
   Google Scholar
• Birkenmeier, G. F., Kim, J. Y., Park, J. K. (2001). Principally quasi-Baer rings. Commun. Algebra 29(2):639–660. DOI: 10.1081/AGB-100001530.
   Web of Science ®Google Scholar
• Birkenmeier, G. F., Park, J. K., Rizvi, S. T. (2013). Extensions of Rings and Modules. New York: Birkhäuser.
   Google Scholar
• Brewer, J., Heinzer, W. (1974). Associated primes of principal ideals. Duke Math. J. 41(1):1–7. DOI: 10.1215/S0012-7094-74-04101-5.
   Google Scholar
• Burdík, Č., Navrátil, O. (2009). Decomposition of the enveloping algebra so(5). In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A., eds. Generalized Lie Theory in Mathematics, Physics and Beyond. Berlin, Heidelberg: Springer, pp. 297–302.
   Google Scholar
• Cohn, P. M. (1999). Reversible rings. Bull. Lond. Math. Soc. 31(6):641–648. DOI: 10.1112/S0024609399006116.
   Web of Science ®Google Scholar
• Faith, C. (1989). Rings with zero intersection property on annihilators: ZIP rings. Publ. Math. 33(2):329–338. DOI: 10.5565/PUBLMAT_33289_09.
   Google Scholar
• Faith, C. (1991). Annihilator ideals, associated primes and Kasch-McCoy commutative rings. Commun. Algebra 19(7):1867–1892.
   Web of Science ®Google Scholar
• Faith, C. (2000). Associated primes in commutative polynomial rings. Commun. Algebra 28(8):3983–3986. DOI: 10.1080/00927870008827069.
   Web of Science ®Google Scholar
• Fajardo, W., Gallego, C., Lezama, O., Reyes, A., Suárez, H., Venegas, H. (2020). Skew PBW Extensions: Ring and Module-theoretic properties, Matrix and Gröbner Methods, and Applications. Algebra and Applications. Cham: Springer.
   Google Scholar
• Gallego, C., Lezama, O. (2011). Gröbner bases for ideals of σ-PBW extensions. Commun. Algebra 39(1):50–75. DOI: 10.1080/00927870903431209.
   Web of Science ®Google Scholar
• Golovashkin, A. V., Maximov, V. M. (2005). On algebras of skew polynomials generated by quadratic homogeneous relations. J. Math. Sci. (N. Y.) 129(2):3757–3771. DOI: 10.1007/s10958-005-0311-z.
   Google Scholar
• Gómez Torrecillas, J. G. (2014). Basic module theory over non-commutative rings with computational aspects of operator algebras. In: Barkatou, M., Cluzeau, T., Regensburger, G., Rosenkranz, M., eds. Algebraic and Algorithmic Aspects of Differential and Integral Operators. AADIOS 2012. Lecture Notes in Computer Science, Vol. 8372. Berlin, Heidelberg: Springer, pp. 23–82.
   Google Scholar
• Green, H. S. (1953). A generalized method of field quantization. Phys. Rev. 90(2):270. DOI: 10.1103/PhysRev.90.270.
   Google Scholar
• Greenberg, O. W., Messiah, A. M. L. (1965). Selection rules for parafields and the absence of para particles in nature. Phys. Rev. 138(5B):B1155–B1167. DOI: 10.1103/PhysRev.138.B1155.
   Google Scholar
• Hamidizadeh, M., Hashemi, E., Reyes, A. (2020). A classification of ring elements in skew PBW extensions over compatible rings. Int. Electron. J. Algebra 28(1):75–97.
   Web of Science ®Google Scholar
• Han, J., Hirano, Y., Kim, H. (2000). Semiprime Ore extensions. Commun. Algebra 28(8):3795–3801. DOI: 10.1080/00927870008827058.
   Web of Science ®Google Scholar
• Hashemi, E., Khalilnezhad, Kh., Alhevaz, A. (2017). (Σ,Δ) -compatible skew PBW extension ring. Kyungpook Math. J. 57(3):401–417.
   Web of Science ®Google Scholar
• Hashemi, E., Khalilnezhad, Kh., Alhevaz, A. (2019). Extensions of rings over 2-primal rings. Matematiche 74(1):141–162.
   Web of Science ®Google Scholar
• Hashemi, E., Moussavi, A., Haj Seyyed Javadi, H. (2003). Polynomial Ore extensions of Baer and p.p.-rings. Bull. Iranian Math. Soc. 29(2):65–86.
   Google Scholar
• Hashemi, E., Moussavi, A. (2005). Polinomial extensions of quasi-Baer rings. Acta Math. Hungar. 107(3):207–224. DOI: 10.1007/s10474-005-0191-1.
   Web of Science ®Google Scholar
• Havlíček, M., Klimyk, A. U., Pošta, S. (2000). Central elements of the algebras U′(som) and U(isom) . Czech. J. Phys. 50(1):79–84.
   Google Scholar
• Hong, C. Y., Kwak, T. K., Rizvi, S. T. (2006). Extensions of generalized Armendariz rings. Algebra Colloq. 13(2):253–266. DOI: 10.1142/S100538670600023X.
   Web of Science ®Google Scholar
• Higuera, S., Reyes, A. (2022). A survey on the fusible property of skew PBW extensions. J. Algebraic Syst. 10(1):1–29.
   Google Scholar
• Hong, C. Y., Kim, N. K., Kwak, T. K. (2000). Ore extensions of Baer and p.p.-rings. J. Pure Appl. Algebra 151(3):215–226. DOI: 10.1016/S0022-4049(99)00020-1.
   Web of Science ®Google Scholar
• Isaev, A. P., Pyatov, P. N., Rittenberg, V. (2001). Diffusion algebras. J. Phys. A. 34(29):5815–5834. DOI: 10.1088/0305-4470/34/29/306.
   Google Scholar
• Jordan, D. A. (1995). Finite-dimensional simple modules over certain iterated skew polynomial rings. J. Pure Appl. Algebra 98(1):45–55. DOI: 10.1016/0022-4049(94)00026-F.
   Web of Science ®Google Scholar
• Jordan, D. A. (2000). Down-up algebras and Ambiskew polynomial rings. J. Algebra 228(1):311–346. DOI: 10.1006/jabr.1999.8264.
   Web of Science ®Google Scholar
• Jordan, D. A., Wells, I. (1996). Invariants for automorphisms of certain iterated skew polynomial rings. Proc. Edinb. Math. Soc. (2) 39(3):461–472. DOI: 10.1017/S0013091500023221.
   Google Scholar
• Kac, V. G. (1977). Lie superalgebras. Adv. Math. 26(1):8–96. DOI: 10.1016/0001-8708(77)90017-2.
   Web of Science ®Google Scholar
• Kanakoglou, K., Daskaloyannis, C. (2009). Bosonisation and parastatistics. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A., eds. Generalized Lie Theory in Mathematics, Physics and Beyond. Berlin, Heidelberg: Springer,pp. 207–218.
   Google Scholar
• Kandri-Rody, A., Weispfenning, V. (1990). Non-commutative Gröbner bases in algebras of solvable type. J. Symbolic Comput. 9(1):1–26. DOI: 10.1016/S0747-7171(08)80003-X.
   Web of Science ®Google Scholar
• Kaplansky, I. (1965). Rings of Operators. Mathematics Lecture Notes Series. New York: Benjamin.
   Google Scholar
• Kirkman, E., Musson, I. M., Passman, D. S. (1999). Noetherian down-up algebras. Proc. Amer. Math. Soc. 127(11):3161–3167. DOI: 10.1090/S0002-9939-99-04926-6.
   Web of Science ®Google Scholar
• Krempa, J. (1996). Some examples of reduced rings. Algebra Colloq. 3(4):289–300.
   Google Scholar
• Kulkarni, R. S. (1999). Irreducible Representations of Witten’s Deformations of U(sl2) . J. Algebra 214(1):64–91.
   Web of Science ®Google Scholar
• Le Bruyn, L. (1994). Two remarks on Witten’s quantum enveloping algebra. Commun. Algebra 22(3):865–876. DOI: 10.1080/00927879408824881.
   Web of Science ®Google Scholar
• Le Bruyn, L. (1995). Conformal sl2 enveloping algebras. Commun. Algebra 23(4):1325–1362.
   Web of Science ®Google Scholar
• Leroy, A., Matczuk, J. (2004). On Induced modules over ore extensions. Commun. Algebra 32(7):2743–2766. DOI: 10.1081/AGB-120037414.
   Web of Science ®Google Scholar
• Lezama, O. (2020). Computation of point modules of finitely semi-graded rings. Commun. Algebra 48(2):866–878. DOI: 10.1080/00927872.2019.1666404.
   Web of Science ®Google Scholar
• Lezama, O., Acosta, J. P., Reyes, A. (2015). Prime ideals of skew PBW extensions. Rev. Un. Mat. Argentina 56(2):39–55.
   Web of Science ®Google Scholar
• Lezama, O., Reyes, A. (2014). Some homological properties of skew PBW extensions. Commun. Algebra 42(3):1200–1230. DOI: 10.1080/00927872.2012.735304.
   Web of Science ®Google Scholar
• Louzari, M., Reyes, A. (2020). Minimal prime ideals of skew PBW extensions over 2-primal compatible rings. Rev. Colombiana Mat. 54(1):39–63. DOI: 10.15446/recolma.v54n1.89788.
   Google Scholar
• Louzari, M., Reyes, A. (2020). Generalized rigid modules and their polynomial extensions. In: Siles Molina, M., El Kaoutit, L., Louzari, M., Ben Yakoub, L., Benslimane, M., eds. Associative and Non-Associative Algebras and Applications. MAMAA 2018. Springer, Proceeding in Mathematics & Statistics, Vol. 311. Cham: Springer,pp. 147–158.
   Google Scholar
• Marks, G. (2001). On 2-primal Ore extensions. Commun. Algebra 29(5):2113–2123. DOI: 10.1081/AGB-100002173.
   Web of Science ®Google Scholar
• Marks, G. (2003). A taxonomy of 2-primal rings. J. Algebra 266(2):494–520. DOI: 10.1016/S0021-8693(03)00301-6.
   Web of Science ®Google Scholar
• McConnell, J., Robson, J. (2001). Noncommutative Noetherian Rings. Graduate Studies in Mathematics. New York: American Mathematical Society.
   Google Scholar
• Macdonald, I. G. (1973). Secondary representation of modules over a commutative ring. Sympos. Math. 11:23–43.
   Google Scholar
• Niño, A., Ramírez, M. C., Reyes, A. (2020). Associated prime ideals over skew PBW extensions. Commun. Algebra 48(12):5038–5055. DOI: 10.1080/00927872.2020.1778012.
   Web of Science ®Google Scholar
• Niño, A., Reyes, A. (2020). Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions. Algebra Discrete Math. 30(2):207–229. DOI: 10.12958/adm1307.
   Web of Science ®Google Scholar
• Ouyang, L., Birkenmeier, G. F. (2012). Weak annihilator over extension rings. Bull. Malays. Math. Sci. Soc. 35(2):345–347.
   Web of Science ®Google Scholar
• Ouyang, L., Liu, J. (2011). On weak (α,δ) -compatible rings. Int. J. Algebra 5(26):1283–1296.
   Google Scholar
• Ouyang, L., Liu, J. (2012). Weak associated primes over differential polynomial rings. Rocky Mountain J. Math. 42(5):1583–1600. DOI: 10.1216/RMJ-2012-42-5-1583.
   Web of Science ®Google Scholar
• Ouyang, L., Liu, J., Xiang, Y. (2013). Ore extensions of skew π-Armendariz rings. Bull. Iranian Math. Soc. 39(2):355–368.
   Web of Science ®Google Scholar
• Ore, O. (1933). Theory of non-commutative polynomials. Ann. Math. (2) 34(3):480–508. DOI: 10.2307/1968173.
   Google Scholar
• Redman, I. T. (1999). The homogenization of the three dimensional skew polynomial algebras of type I. Commun. Algebra 27(11):5587–5602. DOI: 10.1080/00927879908826775.
   Web of Science ®Google Scholar
• Reyes, A. (2018). σ-PBW Extensions of Skew Π-Armendariz rings. Far East J. Math. Sci. (FJMS) 103(2):401–428. DOI: 10.17654/MS103020401.
   Google Scholar
• Reyes, A. (2019). Armendariz modules over skew PBW extensions. Commun. Algebra 47(3):1248–1270. DOI: 10.1080/00927872.2018.1503281.
   Web of Science ®Google Scholar
• Reyes, A., Rodríguez, C. (2021). The McCoy condition on Skew Poincaré-Birkhoff-Witt extensions. Commun. Math. Stat. 9(1):1–21. DOI: 10.1007/s40304-019-00184-5.
   Web of Science ®Google Scholar
• Reyes, A., Sarmiento, C. (2022). On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras. Int. J. Algebra Comput. 32(03):529–559. DOI: 10.1142/S0218196722500242.
   Web of Science ®Google Scholar
• Reyes, A., Suárez, H. (2018). A notion of compatibility for Armendariz and Baer properties over skew PBW extensions. Rev. Un. Mat. Argentina 59(1):157–178.
   Web of Science ®Google Scholar
• Reyes, A., Suárez, H. (2020). Skew Poincaré-Birkhoff-Witt extensions over weak compatible rings. J. Algebra Appl. 19(12):2050225. DOI: 10.1142/S0219498820502254.
   Web of Science ®Google Scholar
• Reyes, A., Suárez, H. (2021). Radicals and Köthe’s conjecture for Skew PBW extensions. Commun. Math. Stat. 9(2):119–138. DOI: 10.1007/s40304-019-00189-0.
   Web of Science ®Google Scholar
• Reyes, A., Suárez, H. (2021). Skew PBW extensions over symmetric rings. Algebra Discrete Math. 32(1):76–102. DOI: 10.12958/adm1767.
   Web of Science ®Google Scholar
• Rosenberg, A. (1995). Non-commutative Algebraic Geometry and Representations of Quantized Algebras. Mathematics and Its Applications, Vol. 330. Dordrecht: Springer.
   Google Scholar
• Shock, R. C. (1972). Polynomial rings over finite dimensional rings. Pacific J. Math. 42(1):251–257. DOI: 10.2140/pjm.1972.42.251.
   Web of Science ®Google Scholar
• Seiler, W. M. (2010). Involution. The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, Vol. 24. Berlin: Springer.
   Google Scholar
• Suárez, H., Chacón, A., Reyes, A. (2022). On NI and NJ skew PBW extensions. Commun. Algebra 50(8):3261–3275. DOI: 10.1080/00927872.2022.2028799.
   Web of Science ®Google Scholar
• Sun, S. H. (1991). Noncommutative rings in which every prime ideal is contained in a unique maximal ideal. J. Pure Appl. Algebra 76(2):179–192. DOI: 10.1016/0022-4049(91)90060-F.
   Web of Science ®Google Scholar
• Tumwesigye, A. B., Richter, J., Silvestrov, S. (2020). Centralizers in PBW extensions. In: Silvestrov, S. Malyarenko, A., Rancic, M., eds. Algebraic Structures and Applications. SPAS 2017. Springer, Proceedings in Mathematics & Statistics, Vol. 317, Springer, pp. 469–490.
   Google Scholar
• Witten, E. (1990). Gauge theories, vertex models, and quantum groups. Nuclear Phys. B 330(2–3):285–346. DOI: 10.1016/0550-3213(90)90115-T.
   Web of Science ®Google Scholar
• Witten, E. (1991). Quantization of Chern-Simons Gauge theory with complex gauge group. Commun. Math. Phys. 137(1):29–66. DOI: 10.1007/BF02099116.
   Web of Science ®Google Scholar
• Woroa̧nowicz, S. L. (1987). Twisted SU(2)-group, an example of a non-commutative differential calculus. Publ. Res. Inst. Math. Sci. 23:117–181.
   Web of Science ®Google Scholar
• Zelmanowitz, J. M. (1976). The finite intersection property on annihilator right ideals. Proc. Amer. Math. Soc. 57(2):213–216. DOI: 10.1090/S0002-9939-1976-0419512-6.
   Web of Science ®Google Scholar
• Zhedanov, A. S. (1991). “Hidden symmetry” of Askey–Wilson polynomials. Theoret. Math. Phys. 89(2):1146–1157. DOI: 10.1007/BF01015906.
   Web of Science ®Google Scholar