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On noncommutative finite factorization domains


Authors: Jason P. Bell, Albert Heinle and Viktor Levandovskyy
Journal: Trans. Amer. Math. Soc. 369 (2017), 2675-2695
MSC (2010): Primary 16U10, 16U30, 16W70
DOI: https://doi.org/10.1090/tran/6727
Published electronically: August 3, 2016
MathSciNet review: 3592524
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Abstract: A domain $ R$ is said to have the finite factorization property if every nonzero nonunit element of $ R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let $ k$ be an algebraically closed field and let $ A$ be a $ k$-algebra. We show that if $ A$ has an associated graded ring that is a domain with the property that the dimension of each homogeneous component is finite, then $ A$ is a finite factorization domain. As a corollary, we show that many classes of algebras have the finite factorization property, including Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, quantum affine spaces and shift algebras. This provides a termination criterion for factorization procedures over these algebras. In addition, we give explicit upper bounds on the number of distinct factorizations of an element in terms of data from the filtration.


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  • [1] Daniel D. Anderson (ed.), Factorization in integral domains, Lecture Notes in Pure and Applied Mathematics, vol. 189, Marcel Dekker, Inc., New York, 1997. MR 1460766 (98c:13002)
  • [2] D. D. Anderson and David F. Anderson, Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80 (1992), no. 3, 217-235. MR 1170712 (93h:13021), https://doi.org/10.1016/0022-4049(92)90144-5
  • [3] D. D. Anderson, David F. Anderson, and Muhammad Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), no. 1, 1-19. MR 1082441 (92b:13028), https://doi.org/10.1016/0022-4049(90)90074-R
  • [4] D. D. Anderson and Bernadette Mullins, Finite factorization domains, Proc. Amer. Math. Soc. 124 (1996), no. 2, 389-396. MR 1322910 (96i:13001), https://doi.org/10.1090/S0002-9939-96-03284-4
  • [5] J. Apel, Gröbnerbasen in nichtkommutativen Algebren und ihre Anwendung, Dissertation, Universität Leipzig (1988).
  • [6] Nicholas R. Baeth and Daniel Smertnig, Factorization theory: from commutative to noncommutative settings, J. Algebra 441 (2015), 475-551. MR 3391936, https://doi.org/10.1016/j.jalgebra.2015.06.007
  • [7] B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Gröbner-bases, Symbolic and algebraic computation (EUROSAM '79, Internat. Sympos., Marseille, 1979) Lecture Notes in Comput. Sci., vol. 72, Springer, Berlin-New York, 1979, pp. 3-21. MR 575678 (82e:14004)
  • [8] J. L. Bueso, J. Gómez-Torrecillas, and F. J. Lobillo, Re-filtering and exactness of the Gelfand-Kirillov dimension, Bull. Sci. Math. 125 (2001), no. 8, 689-715. MR 1872601 (2002k:16045), https://doi.org/10.1016/S0007-4497(01)01090-9
  • [9] José Bueso, José Gómez-Torrecillas, and Alain Verschoren, Algorithmic methods in non-commutative algebra: Applications to quantum groups, Mathematical Modelling: Theory and Applications, vol. 17, Kluwer Academic Publishers, Dordrecht, 2003. MR 2006329 (2005c:16069)
  • [10] P. M. Cohn, Noncommutative unique factorization domains, Trans. Amer. Math. Soc. 109 (1963), 313-331. MR 0155851 (27 #5785)
  • [11] David Cox, John Little, and Donal O'Shea, Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1189133 (93j:13031)
  • [12] David Eisenbud, Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • [13] Mark Giesbrecht, Albert Heinle, and Viktor Levandovskyy, Factoring linear differential operators in $ n$ variables, ISSAC 2014--Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2014, pp. 194-201. MR 3239926, https://doi.org/10.1145/2608628.2608667
  • [14] José Gómez-Torrecillas, Basic module theory over non-commutative rings with computational aspects of operator algebras, Algebraic and algorithmic aspects of differential and integral operators, Lecture Notes in Comput. Sci., vol. 8372, Springer, Heidelberg, 2014, pp. 23-82. With an appendix by V. Levandovskyy. MR 3188538, https://doi.org/10.1007/978-3-642-54479-8_2
  • [15] J. Gómez-Torrecillas and F. J. Lobillo, Auslander-regular and Cohen-Macaulay quantum groups, Algebr. Represent. Theory 7 (2004), no. 1, 35-42. MR 2046952 (2005c:16063), https://doi.org/10.1023/B:ALGE. 0000019384.36800.fa
  • [16] Gert-Martin Greuel and Gerhard Pfister, A SINGULAR introduction to commutative algebra, 2nd, extended edition, with contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann; with 1 CD-ROM (Windows, Macintosh and UNIX), Springer, Berlin, 2008. MR 2363237
  • [17] D. Grigoriev and F. Schwarz, Loewy and primary decompositions of $ \mathcal {D}$-modules, Adv. in Appl. Math. 38 (2007), no. 4, 526-541. MR 2311050 (2008m:32016), https://doi.org/10.1016/j.aam.2005.12.004
  • [18] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57 #3116)
  • [19] A. Kandri-Rody and V. Weispfenning, Noncommutative Gröbner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), no. 1, 1-26. MR 1044911, https://doi.org/10.1016/S0747-7171(08)80003-X
  • [20] Viktor Levandovskyy, Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation, Doctoral Thesis, Universität Kaiserslautern, 2005, http://kluedo.ub.uni-kl.de/volltexte/2005/1883/.
  • [21] Viktor Levandovskyy and Hans Schönemann, PLURAL--a computer algebra system for noncommutative polynomial algebras, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2003, pp. 176-183. MR 2035210, https://doi.org/10.1145/860854.860895
  • [22] Huishi Li, Noncommutative Gröbner bases and filtered-graded transfer, Lecture Notes in Mathematics, vol. 1795, Springer-Verlag, Berlin, 2002. MR 1947291 (2003i:16065)
  • [23] 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
  • [24] Oystein Ore, Theory of non-commutative polynomials, Ann. of Math. (2) 34 (1933), no. 3, 480-508. MR 1503119, https://doi.org/10.2307/1968173
  • [25] Louis Rowen and David J. Saltman, Tensor products of division algebras and fields, J. Algebra 394 (2013), 296-309. MR 3092723, https://doi.org/10.1016/j.jalgebra.2013.07.019
  • [26] Joachim Schmid, On the affine Bezout inequality, Manuscripta Math. 88 (1995), no. 2, 225-232. MR 1354108 (96h:14002), https://doi.org/10.1007/BF02567819
  • [27] Serguei P. Tsarev, An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator, Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 1996, pp. 226-231.

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Additional Information

Jason P. Bell
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: jpbell@uwaterloo.ca

Albert Heinle
Affiliation: David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: aheinle@uwaterloo.ca

Viktor Levandovskyy
Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen University, 52062 Aachen, Germany
Email: viktor.levandovskyy@math.rwth-aachen.de

DOI: https://doi.org/10.1090/tran/6727
Keywords: Finite factorization domains, Weyl algebras, quantum affine spaces, Lie algebras, factorization
Received by editor(s): November 4, 2014
Received by editor(s) in revised form: April 9, 2015, and April 16, 2015
Published electronically: August 3, 2016
Additional Notes: The first two authors thank NSERC for its generous support
Article copyright: © Copyright 2016 American Mathematical Society

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