On free subsemigroups of associative algebras

Letzter, Edward

doi:10.1090/proc/14799

On free subsemigroups of associative algebras
HTML articles powered by AMS MathViewer

by Edward S. Letzter PDF

Proc. Amer. Math. Soc. 148 (2020), 457-460 Request permission

Abstract:

In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative $k$-algebras $R$, where $k$ is a field not algebraic over a finite subfield. We show that $R$ contains a free noncyclic subsemigroup in the following cases: (1) $R$ satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) $R$ has at least one nonartinian primitive subquotient. (3) $k$ is uncountable and $R$ is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Klein’s conjecture for numerous well-known classes of domains, over countable fields, not covered in the prior literature.

References

S. A. Amitsur, A generalization of Hilbert’s Nullstellensatz, Proc. Amer. Math. Soc. 8 (1957), 649–656. MR 87644, DOI 10.1090/S0002-9939-1957-0087644-9
Gwyn Bellamy, Daniel Rogalski, Travis Schedler, J. Toby Stafford, and Michael Wemyss, Noncommutative algebraic geometry, Mathematical Sciences Research Institute Publications, vol. 64, Cambridge University Press, New York, 2016. Lecture notes based on courses given at the Summer Graduate School at the Mathematical Sciences Research Institute (MSRI) held in Berkeley, CA, June 2012. MR 3560285
Amiram Braun, The nilpotency of the radical in a finitely generated PI ring, J. Algebra 89 (1984), no. 2, 375–396. MR 751151, DOI 10.1016/0021-8693(84)90224-2
Ken A. Brown and Ken R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492, DOI 10.1007/978-3-0348-8205-7
Katsuo Chiba, Free subgroups and free subsemigroups of division rings, J. Algebra 184 (1996), no. 2, 570–574. MR 1409229, DOI 10.1006/jabr.1996.0275
Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR 1393197, DOI 10.1090/gsm/011
Vitor O. Ferreira, Érica Z. Fornaroli, and Jairo Z. Gonçalves, Free algebras in division rings with an involution, J. Algebra 509 (2018), 292–306. MR 3812203, DOI 10.1016/j.jalgebra.2018.01.025
Jairo Z. Gonçalves, Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings, Canad. Math. Bull. 27 (1984), no. 3, 365–370. MR 749646, DOI 10.4153/CMB-1984-055-6
J. Z. Gonçalves and M. Shirvani, A survey on free objects in division rings and in division rings with an involution, Comm. Algebra 40 (2012), no. 5, 1704–1723. MR 2924478, DOI 10.1080/00927872.2011.554934
A. A. Klein, Free subsemigroups of domains, Proc. Amer. Math. Soc. 116 (1992), no. 2, 339–341. MR 1096212, DOI 10.1090/S0002-9939-1992-1096212-2
T. Y. Lam, A first course in noncommutative rings, 2nd ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. MR 1838439, DOI 10.1007/978-1-4419-8616-0
A. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 63 (1977), no. 1, 15–16. MR 447432, DOI 10.1090/S0002-9939-1977-0447432-0
A. I. Lichtman, Free subgroups of normal subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 71 (1978), no. 2, 174–178. MR 480623, DOI 10.1090/S0002-9939-1978-0480623-2
L. Makar-Limanov, On free subsemigroups of skew fields, Proc. Amer. Math. Soc. 91 (1984), no. 2, 189–191. MR 740167, DOI 10.1090/S0002-9939-1984-0740167-X
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Revised edition, Graduate Studies in Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2001. With the cooperation of L. W. Small. MR 1811901, DOI 10.1090/gsm/030
D. S. Passman, Universal fields of fractions for polycyclic group algebras, Glasgow Math. J. 23 (1982), no. 2, 103–113. MR 663135, DOI 10.1017/S0017089500004869
Donald S. Passman, The algebraic structure of group rings, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1985. Reprint of the 1977 original. MR 798076
Zinovy Reichstein, On a question of Makar-Limanov, Proc. Amer. Math. Soc. 124 (1996), no. 1, 17–19. MR 1286005, DOI 10.1090/S0002-9939-96-03014-6
Agata Smoktunowicz, Makar-Limanov’s conjecture on free subalgebras, Adv. Math. 222 (2009), no. 6, 2107–2116. MR 2562776, DOI 10.1016/j.aim.2009.07.010
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0