Radicals and Köthe’s Conjecture for Skew PBW Extensions

Abstract

The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincaré–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Köthe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.

Keywords

Armendariz rings / Köthe’s conjecture / Skew PBW extensions

References

[1] Amitsur SA. Algebras over infinite fields. Proc. Am. Math. Soc.. 1956, 7 35-48

[2] Anderson DD, Camillo V. Armendariz rings and Gaussian rings. Commun. Algebra. 1998, 26 7 2265-2275

[3] Artamonov V. Derivations of skew PBW extensions. Commun. Math. Stat.. 2015, 3 4 449-457

[4] Artamonov V, Lezama O, Fajardo W. Extended modules and Ore extensions. Commun. Math. Stat.. 2016, 4 2 189-202

[5] Bell A, Goodearl K. Uniform rank over differential operator rings and Poincaré–Birkhoff–Witt extensions. Pac. J. Math.. 1988, 131 11 13-37

[6] Gallego C, Lezama O. Gröbner bases for ideals of $\sigma $-PBW extensions. Commun. Algebra. 2011, 39 1 50-75

[7] Gallego C, Lezama O. Projective modules and Gröbner bases for skew PBW extensions. Diss. Math.. 2017, 521 1-50

[8] Hinchcliffe, O.: Diffusion algebras. PhD Thesis, University of Sheffield, England (2005)

[9] Isaev AP, Pyatov PN, Rittenberg V. Diffusion algebras. J. Phys. A. 2001, 34 29 5815-5834

[10] Kandri-Rody A, Weispfenning V. Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comput.. 1990, 9 1 1-26

[11] Kirkman E, Kuzmanovich J, Zhang JJ. Shephard–Todd–Chevalley theorem for skew polynomial rings. Algebr. Represent. Theory. 2010, 13 2 127-158

[12] Köthe G. Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständing reduzibel ist. Math. Z.. 1930, 32 161-186

[13] Krempa J. Some examples of reduced rings. Algebra Colloq.. 1996, 3 4 289-300

[14] Lam TY. A First Course in Noncommutative Rings. Graduate Texts in Mathematics. 2001 2 New York: Springer

[15] Lezama O, Acosta JP, Reyes A. Prime ideals of skew PBW extensions. Rev. Un. Mat. Argent.. 2015, 56 2 39-55

[16] Lezama O, Reyes A. Some homological properties of skew PBW extensions. Commun. Algebra. 2014, 42 3 1200-1230

[17] Marks G. A taxonomy of 2-primal rings. J. Algebra. 2003, 266 2 494-520

[18] McConnell JC, Robson JC. Noncommutative Noetherian Rings. Graduate Studies in Mathematics. 2001 Providence: AMS

[19] Nasr-Isfahani AR. Radicals of Ore extension of skew Armendariz rings. Commun. Algebra. 2016, 44 3 1302-1320

[20] Niño D, Reyes A. Some ring theoretical properties for skew Poincaré–Birkhoff–Witt extensions. Bol. Mat. (N. S.). 2017, 24 2 131-148

[21] Ore O. Theory of non-commutative polynomials. Ann. Math.. 1933, 34 3 480-508

[22] Rege MB, Chhawchharia S. Armendariz rings. Proc. Jpn. Acad. Ser. A Math. Sci. 1997, 73 1 14-17

[23] Reyes A. Gelfand–Kirillov dimension of skew PBW extensions. Rev. Colomb. Mat.. 2013, 47 1 95-111

[24] Reyes A. Jacobson’s conjecture and skew PBW extensions. Rev. Integr. Temas Mat.. 2014, 32 2 139-152

[25] Reyes A. Skew PBW extensions of Baer, quasi-Baer, p.p and p.q-rings. Rev. Integr. Temas Mat.. 2015, 33 2 173-189

[26] Reyes A. $\sigma $-PBW extensions of skew $\Pi $-Armendariz rings. Far East J. Math. Sci. (FJMS). 2018, 103 2 401-428

[27] Reyes A. Armendariz modules over skew PBW extensions. Commun. Algebra.. 2019, 47 3 1248-1270

[28] Reyes A, Suárez H. Armendariz property for skew PBW extensions and their classical ring of quotients. Rev. Integr. Temas Mat.. 2016, 34 2 147-168

[29] Reyes A, Suárez H. PBW bases for some 3-dimensional skew polynomial algebras. Far East J. Math. Sci. (FJMS). 2017, 101 6 1207-1228

[30] Reyes A, Suárez H. Bases for quantum algebras and skew Poincaré–Birkhoff–Witt extensions. Momento. 2017, 54 1 54-75

[31] Reyes A, Suárez H. $\sigma $-PBW extensions of skew Armendariz rings. Adv. Appl. Clifford Algebras. 2017, 27 4 3197-3224

[32] Reyes A, Suárez H. Enveloping algebra and skew Calabi–Yau algebras over skew Poincaré–Birkhoff–Witt extensions. Far East J. Math. Sci. (FJMS). 2017, 102 2 373-397

[33] Reyes A, Suárez H. A notion of compatibility for Armendariz and Baer properties over skew PBW extensions. Rev. Un. Mat. Argent.. 2018, 59 1 157-178

[34] Reyes A, Suárez H. Skew Poincaré-Birkhoff-Witt extensions over weak zip rings. Beitr. Algebra Geom.. 2019, 60 2 197-216

[35] Reyes A, Suárez Y. On the ACCP in skew in skew Poincaré-Birkhoff-Witt extensions. Beitr. Algebra Geom.. 2018, 59 4 625-643

[36] Smoktunowicz A. Polynomial rings over nil rings need not be nil. J. Algebra. 2000, 233 2 427-436

[37] Suárez H, Reyes A. A generalized Koszul property for skew PBW extensions. Far East J. Math. Sci. (FJMS). 2017, 101 2 301-320

[38] Suárez Y, Reyes A. Koszulity for skew PBW extensions over fields. JP J. Algebra Number Theory Appl.. 2017, 39 2 181-203

Funding

Universidad Nacional de Colombia