Prime and primitive algebras with prescribed growth types. (English) Zbl 1371.16019
In [Q. J. Math. 65, No. 2, 421–438 (2014; Zbl 1312.16017)], A. Smoktunowicz and L. Bartholdi constructed finitely generated monomial algebras with prescribed sufficiently fast growth types. The authors showed that their construction does not necessarily result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type. Using a construction of an inverse system of monomial ideals which arise from this construction, the author presents the construction of finitely generated primitive algebras without further limitations on the growth type. The author is inspired by E. I. Zel’manov’s example in [Sib. Math. J. 20, 303–304 (1979; Zbl 0433.16005)], so in this paper, he studies how prime algebras can be constructed such that they contain non-zero locally nilpotent ideals; this is the very opposite of primitive constructions. In fact, the reader will find good results about this area of algebras, where the author discusses deeply this problem, and explains some results by examples about non-semiprime algebras.
Moreover, the author closes this paper with two open questions, which are: What are the possible growth types of finitely generated simple algebras? What are the possible growth types of finitely generated domain?
Moreover, the author closes this paper with two open questions, which are: What are the possible growth types of finitely generated simple algebras? What are the possible growth types of finitely generated domain?
Reviewer: Mehsin Atteya (Leicester)
References:
| [1] | L. Bartholdi and A. Smoktunowicz, Images of Golod-Shafarevich algebras with small growth, Quarterly Journal of Mathematics 65 (2014), 421-438. · Zbl 1312.16017 · doi:10.1093/qmath/hat005 |
| [2] | K. I. Beidar and Y. Fong, On radicals of monomial algebras, Communications in Algebra 26 (1998), 3913-3919. · Zbl 0916.16013 · doi:10.1080/00927879808826384 |
| [3] | G. M. Bergman, On Jacobson radicals of graded rings, unpublished (1975), http://math.berkeley.edu/ gbergman/papers/unpub/J_G.pdf. |
| [4] | J. Cassaigne, Constructing innite words of intermediate complexity, in Developments in Language Theory, Lecture Notes in Computer Science, Vol. 2450, Springer, Berlin, 2003, pp. 173-184. · Zbl 1015.68138 · doi:10.1007/3-540-45005-X_15 |
| [5] | P. De La Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. · Zbl 0965.20025 |
| [6] | R. I. Grigorchuk, Degrees of growth of p-groups and torsion-free groups, Matematicheskiĭ Sbornik 126 (168) (1985), 194-214. · Zbl 0568.20033 |
| [7] | Grigorchuk, R., On growth in group theory, 325-338 (1991) · Zbl 0749.20016 |
| [8] | R. Grigorchuk, Milnor’s problem on the growth of groups and its consequences, in Frontiers in Complex Dynamics, Princeton Mathematical Series, Vol. 51, Princeton University Press, Princeton, NJ, 2014, pp. 705-773. · Zbl 1325.20034 |
| [9] | V. Nekrashevych, Growth of étale groupoids and simple algebras, International Journal of Algebra and Computation 26 (2016), 375-397. · Zbl 1366.16016 · doi:10.1142/S0218196716500156 |
| [10] | J. Okninski, Structure of prime finitely presented monomial algebras, Journal of Algebra 320 (2008), 3199-3205. · Zbl 1196.16025 · doi:10.1016/j.jalgebra.2008.08.003 |
| [11] | D. S. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sidney, 1977. · Zbl 0368.16003 |
| [12] | M. K. Smith, Universal enveloping algebras with subexponenetial but not polynomially bounded growth, Proceedings of the American Mathematical Society 60 (1976), 22-24. · Zbl 0347.17005 · doi:10.1090/S0002-9939-1976-0419534-5 |
| [13] | A. Smoktunowicz, Growth, entropy and commutativity of algebras satisfying prescribed relations, Selecta Mathematica 20 (2014), 1197-1212. · Zbl 1320.16009 · doi:10.1007/s00029-014-0154-x |
| [14] | A. Smoktunowicz and U. Vishne, An affine prime non-semiprimitive monomial algebra with quadratic growth, Advances in Applied Mathematics 37 (2006), 511-513. · Zbl 1126.16013 · doi:10.1016/j.aam.2005.03.008 |
| [15] | V. I. Trofimov, The growth functions of finitely generated semigroups, Semigroup Forum 21 (1980), 351-360. · Zbl 0453.20047 · doi:10.1007/BF02572559 |
| [16] | U. Vishne, Primitive algebras with arbitrary Gelfand-Kirillov dimension, Journal of Algebra 211 (1999), 151-158. · Zbl 0926.16024 · doi:10.1006/jabr.1998.7567 |
| [17] | E. Zelmanov, An example of a finitely generated prime ring, Siberian Mathematical Journal 20 (1979), 303-304. · Zbl 0433.16005 · doi:10.1007/BF00970042 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
