Skip to main content
SpringerLink
Log in

Factorizations of Elements in Noncommutative Rings: A Survey

  • Conference paper
  • First Online:
Multiplicative Ideal Theory and Factorization Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 170))

Abstract

We survey results on factorizations of non-zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of nonunique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of nonunique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields .

Dedicated to Franz Halter-Koch on the occasion of his 70th birthday

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We may always force this condition by replacing H by the subcategory of all cancellative elements. Note that then principal right ideals aH have to be replaced by \(aH^\bullet \). Sometimes it can be more convenient work with H with \(H^\bullet \ne H\), because typically we will have \(H=R\) a ring and \(H^\bullet = R^\bullet \) the semigroup of non-zero-divisors. In this setting, sufficient conditions for the stated condition to be satisfied are for \(R^\bullet \) to be Ore, or R to be a domainWe may always force this condition by replacing H by the subcategory of all cancellative elements. Note that then principal right ideals aH have to be replaced by \(aH^\bullet \). Sometimes it can be more convenient work with H with \(H^\bullet \ne H\), because typically we will have \(H=R\) a ring and \(H^\bullet = R^\bullet \) the semigroup of non-zero-divisors. In this setting, sufficient conditions for the stated condition to be satisfied are for \(R^\bullet \) to be Ore, or R to be a domain.

References

  1. G.Q. Abbasi, S. Kobayashi, H. Marubayashi, A. Ueda, Noncommutative unique factorization rings. Commun. Algebra 19(1), 167–198 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Asano, K. Murata, Arithmetical ideal theory in semigroups. J. Inst. Polytech. Osaka City Univ. Ser. A. Math. 4, 9–33 (1953)

    Google Scholar 

  3. E. Akalan, H. Marubayashi, Multiplicative ideal theory in non-commutative rings, in Multiplicative Ideal Theory and Factorization Theory (Springer, Berlin, 2016) preprint

    Google Scholar 

  4. D.D. Anderson (ed.), Factorization in Integral Domains, vol. 189. Lecture Notes in Pure and Applied Mathematics (Marcel Dekker Inc, New York, 1997)

    Google Scholar 

  5. D. Bachman, N.R. Baeth, J. Gossell, Factorizations of upper triangular matrices. Linear Algebra Appl. 450, 138–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. R.A. Beauregard, Right \({\rm{LCM}}\) domains. Proc. Am. Math. Soc. 30, 1–7 (1971)

    MathSciNet  MATH  Google Scholar 

  7. R.A. Beauregard, Right-bounded factors in an LCM domain. Trans. Am. Math. Soc. 200, 251–266 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  8. R.A. Beauregard, An analog of Nagata’s theorem for modular LCM domains. Can. J. Math. 29(2), 307–314 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  9. R.A. Beauregard, Left versus right LCM domains. Proc. Am. Math. Soc. 78(4), 464–466 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. R.A. Beauregard, Unique factorization in the ring \(R[x]\). J. Aust. Math. Soc. Ser. A 53(3), 287–293 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. R.A. Beauregard, When is \(F[x, y]\) a unique factorization domain? Proc. Am. Math. Soc. 117(1), 67–70 (1993)

    MathSciNet  MATH  Google Scholar 

  12. R.A. Beauregard, Factorization in LCM domains with conjugation. Can. Math. Bull. 37(3), 289–293 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. R.A. Beauregard, Right unique factorization domains. Rocky Mountain J. Math. 24(2), 483–489 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. R.A. Beauregard, Rings with the right atomic multiple property. Commun. Algebra 23(3), 1017–1026 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. G.M. Bergman, Commuting elements in free algebras and related topics in ring theory. Ph.D. thesis, Harvard University, 1968

    Google Scholar 

  16. J. Bergen, M. Giesbrecht, P.N. Shivakumar, Y. Zhang, Factorizations for difference operators. Adv. Differ. Equ. 57, 6 (2015)

    MathSciNet  Google Scholar 

  17. J.P. Bell, A. Heinle, V. Levandovskyy, On noncommutative finite factorization domains. Trans. Am. Math. Soc. (2015) to appear

    Google Scholar 

  18. A.J. Berrick, M.E. Keating, An Introduction to Rings and Modules with K-Theory in View, vol. 65. Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  19. N.R. Baeth, V. Ponomarenko, D. Adams, R. Ardila, D. Hannasch, A. Kosh, H. McCarthy, R. Rosenbaum, Number theory of matrix semigroups. Linear Algebra Appl. 434(3), 694–711 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. K.A. Brown, Height one primes of polycyclic group rings. J. Lond. Math. Soc. (2) 32(3), 426–438 (1985)

    Google Scholar 

  21. H.-H. Brungs, Ringe mit eindeutiger Faktorzerlegung. J. Reine Angew. Math. 236, 43–66 (1969)

    MathSciNet  MATH  Google Scholar 

  22. N.R. Baeth, D. Smertnig, Factorization theory: from commutative to noncommutative settings. J. Algebra 441, 475–551 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. A.W. Chatters, J. Clark, Group rings which are unique factorisation rings. Commun. Algebra 19(2), 585–598 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. A.W. Chatters, M.P. Gilchrist, D. Wilson, Unique factorisation rings. Proc. Edinb. Math. Soc. (2), 35(2), 255–269 (1992)

    Google Scholar 

  25. M. Chamarie, Anneaux de Krull non commutatifs. J. Algebra 72(1), 210–222 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. A.W. Chatters, Noncommutative unique factorization domains. Math. Proc. Camb. Philos. Soc. 95(1), 49–54 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. A.W. Chatters, Unique factorisation in P.I. group-rings. J. Aust. Math. Soc. Ser. A 59(2), 232–243 (1995)

    Google Scholar 

  28. S.T. Chapman (ed.), Arithmetical Properties of Commutative Rings and Monoids, vol. 241. Lecture Notes in Pure and Applied Mathematics (Chapman & Hall/CRC, Boca Raton, 2005)

    Google Scholar 

  29. A.W. Chatters, D.A. Jordan, Noncommutative unique factorisation rings. J. Lond. Math. Soc. (2), 33(1), 22–32 (1986)

    Google Scholar 

  30. H. Cohn, A. Kumar, Metacommutation of Hurwitz primes. Proc. Am. Math. Soc. 143(4), 1459–1469 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. P.M. Cohn, Factorization in non-commutative power series rings. Proc. Camb. Philos. Soc. 58, 452–464 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  32. P.M. Cohn, Noncommutative unique factorization domains. Trans. Am. Math. Soc. 109, 313–331 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  33. P.M. Cohn, Rings with a weak algorithm. Trans. Am. Math. Soc. 109, 332–356 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  34. P.M. Cohn, Errata to: noncommutative unique factorization domains. Trans. Am. Math. Soc. 119, 552 (1965)

    Article  MathSciNet  Google Scholar 

  35. P.M. Cohn, Factorization in general rings and strictly cyclic modules. J. Reine Angew. Math. 239(240), 185–200 (1969)

    MathSciNet  MATH  Google Scholar 

  36. P.M. Cohn, Correction to: unique factorization domains (Am. Math. Mon. 80, 1–18). Am. Math. Mon. 80(1115), 1973 (1973)

    Google Scholar 

  37. P.M. Cohn, Unique factorization domains. Am. Math. Mon. 80, 1–18 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  38. P.M. Cohn, Free Rings and their Relations, vol. 19, 2nd edn. London Mathematical Society Monographs (Academic Press Inc., Harcourt Brace Jovanovich Publishers, London, 1985)

    Google Scholar 

  39. P.M. Cohn, Free Ideal Rings and Localization in General Rings, vol. 3, New Mathematical Monographs (Cambridge University Press, Cambridge, 2006)

    Google Scholar 

  40. C.W. Curtis, I. Reiner, Methods of Representation Theory, vol. II. Pure and Applied Mathematics (Wiley, New York, 1987) (With applications to finite groups and orders, A Wiley-Interscience Publication)

    Google Scholar 

  41. P.M. Cohn, A.H. Schofield, Two examples of principal ideal domains. Bull. Lond. Math. Soc. 17(1), 25–28 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  42. J.H. Conway, D.A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry (A K Peters Ltd., Natick, 2003)

    MATH  Google Scholar 

  43. J. Delenclos, A. Leroy, Noncommutative symmetric functions and \(W\)-polynomials. J. Algebra Appl. 6(5), 815–837 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. D.R. Estes, J.R. Matijevic, Matrix factorizations, exterior powers, and complete intersections. J. Algebra 58(1), 117–135 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  45. D.R. Estes, J.R. Matijevic, Unique factorization of matrices and Towber rings. J. Algebra 59(2), 387–394 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  46. D.R. Estes, G. Nipp, Factorization in quaternion orders. J. Number Theory 33(2), 224–236 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  47. D.R. Estes, Factorization in hereditary orders. Linear Algebra Appl. 157, 161–164 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  48. D.R. Estes, Factorization in quaternion orders over number fields, in The Mathematical Heritage of C. F. Gauss (World Scientific Publishing, River Edge, 1991), pp. 195–203

    Google Scholar 

  49. H. Fitting, Über den Zusammenhang zwischen dem Begriff der Gleichartigkeit zweier Ideale und dem Äquivalenzbegriff der Elementarteilertheorie. Math. Ann. 112(1), 572–582 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Geroldinger, Additive group theory and non-unique factorizations, in Combinatorial Number Theory and Additive Group Theory. Advanced Courses in Mathematics. CRM Barcelona (Birkhäuser Verlag, Basel, 2009), pp. 1–86

    Google Scholar 

  51. A. Geroldinger, Non-commutative Krull monoids: a divisor theoretic approach and their arithmetic. Osaka J. Math. 50(2), 503–539 (2013)

    MathSciNet  MATH  Google Scholar 

  52. I. Gelfand, S. Gelfand, V. Retakh, R.L. Wilson, Quasideterminants. Adv. Math. 193(1), 56–141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Geroldinger, F. Halter-Koch, Non-unique factorizations. Algebraic, combinatorial and analytic theory, vol. 278. Pure and Applied Mathematics (Boca Raton) (Chapman & Hall/CRC, Boca Raton, 2006)

    Google Scholar 

  54. G. Grätzer, J.B. Nation, A new look at the Jordan-Hölder theorem for semimodular lattices. Algebra Universalis 64(3–4), 309–311 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. M.P. Gilchrist, M.K. Smith, Noncommutative UFDs are often PIDs. Math. Proc. Camb. Philos. Soc. 95(3), 417–419 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  56. K.R. Goodearl, R.B. Warfield Jr., An Introduction to Noncommutative Noetherian Rings, vol. 61, 2nd edn. London Mathematical Society Student Texts (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  57. K.R. Goodearl, M.T. Yakimov, From quantum Ore extensions to quantum tori via noncommutative UFDs (2012) Preprint

    Google Scholar 

  58. K.R. Goodearl, M.T. Yakimov, Quantum cluster algebras and quantum nilpotent algebras. Proc. Natl. Acad. Sci. USA 111(27), 9696–9703 (2014)

    Article  MathSciNet  Google Scholar 

  59. A. Heinle, V. Levandovskyy, Factorization of \({z}\)-homogeneous polynomials in the first (\(q\))-Weyl Algebra (2014) preprint

    Google Scholar 

  60. G. Hegedüs, Z. Li, J. Schicho, H.-P. Schröcker, From the fundamental theorem of algebra to Kempe’s universality theorem. Int. Math. Nachr. 229, 13–26 (2015)

    Google Scholar 

  61. E. Hallouin, C. Maire, Cancellation in totally definite quaternion algebras. J. Reine Angew. Math. 595, 189–213 (2006)

    MathSciNet  MATH  Google Scholar 

  62. D. Haile, L.H. Rowen, Factorizations of polynomials over division algebras. Algebra Colloquium 2(2), 145–156 (1995)

    MathSciNet  MATH  Google Scholar 

  63. G. Hegedüs, J. Schicho, H.-P. Schröcker, Construction of overconstrained linkages by factorization of rational motions, in Latest Advances in Robot Kinematics, ed. by J. Lenarcic, M. Husty (Springer, Netherlands, 2012), pp. 213–220

    Chapter  Google Scholar 

  64. G. Hegedüs, J. Schicho, H.-P. Schröcker, Factorization of rational curves in the study quadric. Mech. Mach. Theory 69(1), 142–152 (2013)

    Article  Google Scholar 

  65. N. Jacobson, The Theory of Rings. American Mathematical Society Mathematical Surveys, vol. I (American Mathematical Society, New York, 1943)

    Google Scholar 

  66. E. Jespers, J. Okniński, Noetherian Semigroup Algebras, vol. 7. Algebras and Applications (Springer, Dordrecht, 2007)

    Google Scholar 

  67. D.A. Jordan, Unique factorisation of normal elements in noncommutative rings. Glasgow Math. J. 31(1), 103–113 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  68. E. Jespers, Q. Wang, Noetherian unique factorization semigroup algebras. Commun. Algebra 29(12), 5701–5715 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  69. E. Jespers, Q. Wang, Height-one prime ideals in semigroup algebras satisfying a polynomial identity. J. Algebra 248(1), 118–131 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  70. T.Y. Lam, Serre’s Problem on Projective Modules. Springer Monographs in Mathematics (Springer, Berlin, 2006)

    Google Scholar 

  71. E. Landau, Ein Satz über die Zerlegung homogener linearer Differentialausdrücke in irreductible Factoren. J. Reine Angew. Math. 124, 115–120 (1902)

    MathSciNet  Google Scholar 

  72. L. Le Bruyn, Trace rings of generic matrices are unique factorization domains. Glasgow Math. J. 28(1), 11–13 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  73. A. Leroy, Noncommutative polynomial maps. J. Algebra Appl. 11(4), 1250076, 16 (2012)

    Google Scholar 

  74. D. Lissner, A. Geramita, Towber rings. J. Algebra 15, 13–40 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  75. T.Y. Lam, A. Leroy, Algebraic conjugacy classes and skew polynomial rings, in Perspectives in Ring Theory (Antwerp, 1987), vol. 233. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences (Kluwer Academic Publishers, Dordrecht, 1988), pp. 153–203

    Google Scholar 

  76. T.Y. Lam, A. Leroy, Principal one-sided ideals in Ore polynomial rings, in Algebra and Its Applications (Athens, OH, 1999), vol. 259. Contemporary Mathematics (American Mathematical Society, Providence, 2000), pp. 333–352

    Google Scholar 

  77. T.Y. Lam, A. Leroy, Wedderburn polynomials over division rings. I. J. Pure Appl. Algebra 186(1), 43–76 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  78. T.Y. Lam, A. Leroy, A. Ozturk, Wedderburn polynomials over division rings. II, in Noncommutative Rings, Group Rings, Diagram Algebras and their Applications, vol. 456. Contemporary Mathematics (American Mathematical Society, Providence, RI, 2008), pp. 73–98

    Google Scholar 

  79. S. Launois, T.H. Lenagan, L. Rigal, Quantum unique factorisation domains. J. Lond. Math. Soc. (2) 74(2), 321–340 (2006)

    Google Scholar 

  80. A. Leroy, A. Ozturk, Algebraic and \(F\)-independent sets in 2-firs. Commun. Algebra 32(5), 1763–1792 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  81. A. Loewy, Über reduzible lineare homogene Differentialgleichungen. Math. Ann. 56(4), 549–584 (1903)

    Article  MathSciNet  MATH  Google Scholar 

  82. L.S. Levy, J.C. Robson, Hereditary Noetherian Prime Rings and Idealizers, vol. 174. Mathematical Surveys and Monographs (American Mathematical Society, Providence, 2011)

    Google Scholar 

  83. Z. Li, T.-D. Rad, J. Schicho, H.-P. Schröcker, Factorization of rational motions: a survey with examples and applications, in Proceedings of the 14th IFToMM World Congress (Taipei, 2015) (2015) preprint

    Google Scholar 

  84. H. Marubayashi, Ore extensions over total valuation rings. Algebras Represent. Theory 13(5), 607–622 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  85. J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, vol. 30. With the cooperation of L. W. Small, revised edition. Graduate Studies in Mathematics (American Mathematical Society, Providence, 2001)

    Google Scholar 

  86. C. Maclachlan, A.W. Reid, The Arithmetic of Hyperbolic 3-Manifolds, vol. 219. Graduate Texts in Mathematics (Springer, New York, 2003)

    Google Scholar 

  87. H. Marubayashi, F. Van Oystaeyen, Prime Divisors and Noncommutative Valuation Theory, vol. 2059. Lecture Notes in Mathematics (Springer, Heidelberg, 2012)

    Google Scholar 

  88. W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd edn. Springer Monographs in Mathematics (Springer, Berlin, 2004)

    Google Scholar 

  89. O. Ore, Theory of non-commutative polynomials. Ann. Math. (2), 34(3), 480–508 (1933)

    Google Scholar 

  90. I. Reiner, Maximal Orders. London Mathematical Society Monographs, No. 5 (Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers), London-New York, 1975)

    Google Scholar 

  91. V. Retakh, From factorizations of noncommutative polynomials to combinatorial topology. Central Eur. J. Math. 8(2), 235–243 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  92. D. Smertnig, A note on cancellation in totally definite quaternion algebras. J. Reine Angew. Math. 707, 209–216 (2015)

    Google Scholar 

  93. D. Smertnig, Sets of lengths in maximal orders in central simple algebras. J. Algebra 390, 1–43 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  94. D. Smertnig. Factorizations in bounded hereditary Noetherian prime rings (2016) preprint http://arxiv.org/abs/1605.09274

  95. R.G. Swan, Strong approximation and locally free modules, in Ring Theory and Algebra, III (Proceedings of Third Conference, University Oklahoma, Norman, Okla., 1979), vol. 55. Lecture Notes in Pure and Applied Mathematics (Dekker, New York, 1980), pp. 153–223

    Google Scholar 

  96. J. Towber, Complete reducibility in exterior algebras over free modules. J. Algebra 10, 299–309 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  97. M.-F. Vignéras, Simplification pour les ordres des corps de quaternions totalement définis. J. Reine Angew. Math. 286(287), 257–277 (1976)

    MathSciNet  MATH  Google Scholar 

  98. M. Zieve, P. Müller, On Ritt’s polynomial decomposition theorems (2008) preprint

    Google Scholar 

Download references

Acknowledgments

I thank the anonymous referee for his careful reading. The author was supported by the Austrian Science Fund (FWF) project P26036-N26.

Author information

Authors and Affiliations

  1. Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstrae 36, 8010, Graz, Austria

    Daniel Smertnig

Authors
  1. Daniel Smertnig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Smertnig .

Editor information

Editors and Affiliations

  1. Dept of Math & Stat, Sam Houston State University, Huntsville, Texas, USA

    Scott Chapman

  2. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy

    Marco Fontana

  3. Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, Graz, Austria

    Alfred Geroldinger

  4. Dept. Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico, USA

    Bruce Olberding

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Smertnig, D. (2016). Factorizations of Elements in Noncommutative Rings: A Survey. In: Chapman, S., Fontana, M., Geroldinger, A., Olberding, B. (eds) Multiplicative Ideal Theory and Factorization Theory. Springer Proceedings in Mathematics & Statistics, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-319-38855-7_15

Download citation

Publish with us

Policies and ethics

Search

Navigation