Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Multiplicatively collapsing and rewritable algebras


Authors: Eric Jespers, David Riley and Mayada Shahada
Journal: Proc. Amer. Math. Soc. 143 (2015), 4223-4236
MSC (2010): Primary 16R99, 17B30, 20M25
DOI: https://doi.org/10.1090/S0002-9939-2015-12563-4
Published electronically: March 18, 2015
MathSciNet review: 3373922
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A semigroup $ S$ is called $ n$-collapsing if, for every $ a_1,\ldots , a_n$ in $ S$, there exist functions $ f\neq g$ (depending on $ a_1,\ldots , a_n$), such that

$\displaystyle a_{f(1)}\cdots a_{f(n)} = a_{g(1)}\cdots a_{g(n)};$

it is called collapsing if it is $ n$-collapsing, for some $ n$. More specifically, $ S$ is called $ n$-rewritable if $ f$ and $ g$ can be taken to be permutations; $ S$ is called rewritable if it is $ n$-rewritable for some $ n$. Semple and Shalev extended Zelmanov's solution of the restricted Burnside problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent. In this paper, we consider when the multiplicative semigroup of an associative algebra is collapsing; in particular, we prove the following conditions are equivalent, for all unital algebras $ A$ over an infinite field: the multiplicative semigroup of $ A$ is collapsing, $ A$ satisfies a multiplicative semigroup identity, and $ A$ satisfies an Engel identity. We deduce that, if the multiplicative semigroup of $ A$ is rewritable, then $ A$ must be commutative.

References [Enhancements On Off] (What's this?)

  • [1] Bernhard Amberg and Yaroslav Sysak, Associative rings whose adjoint semigroup is locally nilpotent, Arch. Math. (Basel) 76 (2001), no. 6, 426-435. MR 1831498 (2002b:16029), https://doi.org/10.1007/PL00000453
  • [2] Yuly Billig, David Riley, and Vladimir Tasić, Nonmatrix varieties and nil-generated algebras whose units satisfy a group identity, J. Algebra 190 (1997), no. 1, 241-252. MR 1442155 (98g:16030), https://doi.org/10.1006/jabr.1996.6892
  • [3] Russell D. Blyth, Rewriting products of group elements. II, J. Algebra 119 (1988), no. 1, 246-259. MR 971358 (90b:20034), https://doi.org/10.1016/0021-8693(88)90088-9
  • [4] M. I. Elashiry and D. S. Passman, Rewritable groups, J. Algebra 345 (2011), 190-201. MR 2842061 (2012j:20089), https://doi.org/10.1016/j.jalgebra.2011.08.007
  • [5] O. B. Finogenova, Varieties of associative algebras satisfying Engel identities, Algebra Logika 43 (2004), no. 4, 482-505, 508 (Russian, with Russian summary); English transl., Algebra Logic 43 (2004), no. 4, 271-284. MR 2105850 (2005h:08007), https://doi.org/10.1023/B:ALLO.0000035118.51742.41
  • [6] P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595-622. MR 0110750 (22 #1618)
  • [7] A. R. Kemer, Nonmatrix varieties, Algebra i Logika 19 (1980), no. 3, 255-283, 382 (Russian). MR 609015 (82k:16022)
  • [8] Irving Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575-580. MR 0025451 (10,7a)
  • [9] S. I. Kublanovskii, On varieties of associative algebras with local finiteness conditions, Algebra i Analiz 9 (1997), no. 4, 119-174 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 4, 763-813. MR 1604008 (99a:16017)
  • [10] A. I. Malcev, Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uc. Zap. Fiz.-Mat. Nauki 4 (1953), 107-111 (Russian). MR 0075959 (17,825d)
  • [11] Ju. N. Malcev, Varieties of associative algebras, Algebra i Logika 15 (1976), no. 5, 579-584, 606 (Russian). MR 0485632 (58 #5457)
  • [12] S. P. Mishchenko, V. M. Petrogradsky, and A. Regev, Characterization of non-matrix varieties of associative algebras, Israel J. Math. 182 (2011), 337-348. MR 2783976 (2012e:16057), https://doi.org/10.1007/s11856-011-0034-4
  • [13] B. H. Neumann and Tekla Taylor, Subsemigroups of nilpotent groups, Proc. Roy. Soc. Ser. A 274 (1963), 1-4. MR 0159884 (28 #3100)
  • [14] David Michael Riley, Algebras with collapsing monomials, Bull. London Math. Soc. 30 (1998), no. 5, 521-528. MR 1643826 (99g:16029), https://doi.org/10.1112/S0024609398004597
  • [15] David M. Riley, Algebras with collapsing monomials. II, Comm. Algebra 29 (2001), no. 7, 2745-2756. MR 1848379 (2002e:16048), https://doi.org/10.1081/AGB-100104984
  • [16] David M. Riley, Engel varieties of associative rings and the number of Mersenne primes, J. Algebra 261 (2003), no. 1, 19-30. MR 1967154 (2004b:16032), https://doi.org/10.1016/S0021-8693(02)00562-8
  • [17] D. M. Riley and Mark C. Wilson, Associative rings satisfying the Engel condition, Proc. Amer. Math. Soc. 127 (1999), no. 4, 973-976. MR 1473677 (99f:16025), https://doi.org/10.1090/S0002-9939-99-04643-2
  • [18] David M. Riley and Mark C. Wilson, Associative algebras satisfying a semigroup identity, Glasg. Math. J. 41 (1999), no. 3, 453-462. MR 1720410 (2000j:16037), https://doi.org/10.1017/S0017089599000142
  • [19] James F. Semple and Aner Shalev, Combinatorial conditions in residually finite groups. I, J. Algebra 157 (1993), no. 1, 43-50. MR 1219657 (94g:20033), https://doi.org/10.1006/jabr.1993.1089
  • [20] Aner Shalev, Combinatorial conditions in residually finite groups. II, J. Algebra 157 (1993), no. 1, 51-62. MR 1219658 (94g:20034), https://doi.org/10.1006/jabr.1993.1090
  • [21] Moss E. Sweedler, Lie nilpotent and solvable associative algebras, J. Algebra 26 (1973), 422-430. MR 0379604 (52 #509)
  • [22] E. I. Zelmanov, On Engel Lie algebras, Sibirsk. Mat. Zh. 29 (1988), no. 5, 112-117, 238 (Russian); English transl., Siberian Math. J. 29 (1988), no. 5, 777-781 (1989). MR 971234 (90a:17010), https://doi.org/10.1007/BF00970273
  • [23] E. I. Zelmanov, Solution of the restricted Burnside problem for groups of odd exponent, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 42-59, 221 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 41-60. MR 1044047 (91i:20037)
  • [24] E. I. Zelmanov, Solution of the restricted Burnside problem for $ 2$-groups, Mat. Sb. 182 (1991), no. 4, 568-592 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 543-565. MR 1119009 (93a:20063)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16R99, 17B30, 20M25

Retrieve articles in all journals with MSC (2010): 16R99, 17B30, 20M25


Additional Information

Eric Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: efjesper@vub.ac.be

David Riley
Affiliation: Department of Mathematics, Western University, London, Ontario, Canada N6A 5B7
Email: dmriley@uwo.ca

Mayada Shahada
Affiliation: Department of Mathematics, Western University, London, Ontario, Canada N6A 5B7
Email: mshahada@uwo.ca

DOI: https://doi.org/10.1090/S0002-9939-2015-12563-4
Keywords: Multiplicative semigroup, adjoint semigroup, collapsing, rewritable, semigroup identity, Lie nilpotent, Engel identity
Received by editor(s): March 28, 2014
Received by editor(s) in revised form: June 12, 2014
Published electronically: March 18, 2015
Additional Notes: The authors acknowledge support from Onderzoeksraad of Vrije Universiteit, Fonds voor Wetenschappelijk Onderzoek (Vlaanderen) and NSERC of Canada.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society