Research Article
BibTex RIS Cite
Year 2020, , 9 - 42, 14.07.2020
https://doi.org/10.24330/ieja.768114

Abstract

References

  • N. R. Baeth and D. Smertnig, Factorization theory: from commutative to non-commutative settings, J. Algebra, 441 (2015), 475-551.
  • H. Bart, I. Gohberg, M. A. Kaashoek and A. C. M. Ran, Factorization of Matrix and Operator Functions: the State Space Method, Operator Theory: Advances and Applications, 178, Linear Operators and Linear Systems, Birkhauser Verlag, Basel, 2008.
  • J. Berstel and C. Reutenauer, Noncommutative Rational Series with Applications, Encyclopedia of Mathematics and its Applications, 137, Cambridge University Press, Cambridge, 2011.
  • P. M. Cohn, Noncommutative unique factorization domains, Trans. Amer. Math. Soc., 109 (1963), 313-331.
  • P. M. Cohn, Skew Fields, Theory of General Division Rings, Encyclopedia of Mathematics and its Applications, 57, Cambridge University Press, Cambridge, 1995.
  • P. M. Cohn, Basic Algebra, Groups, Rings and Fields, Springer-Verlag London, Ltd., London, 2003.
  • P. M. Cohn, Further Algebra and Applications, Springer-Verlag London, Ltd., London, 2003.
  • P. M. Cohn, Free Ideal Rings and Localization in General Rings, New Mathematical Monographs, 3, Cambridge University Press, Cambridge, 2006.
  • P. M. Cohn and C. Reutenauer, A normal form in free fields, Canad. J. Math., 46(3) (1994), 517-531.
  • P. M. Cohn and C. Reutenauer, On the construction of the free eld, Dedicated to the memory of Marcel-Paul Schutzenberger, Internat. J. Algebra Comput., 9(3-4) (1999), 307-323.
  • D. A. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Fourth edition, Undergraduate Texts in Mathematics, Springer, Cham, 2015.
  • FriCAS Computer Algebra System, W. Hebisch, 2018. http://axiom-wiki.newsynthesis.org/FrontPage.
  • J. W. Helton, I. Klep and J. Volcic, Geometry of free loci and factorization of noncommutative polynomials, Adv. Math., 331 (2018), 589-626.
  • B. Janko, Factorization of Non-commutative Polynomials and Testing Fullness of Matrices, Diplomarbeit, TU Graz, 2018.
  • D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting, Linear Algebra Appl., 430(4) (2009), 869-889.
  • O. Ore, Linear equations in non-commutative fields, Ann. of Math., 32(3) (1931), 463-477.
  • C. Reutenauer, Michel Fliess and non-commutative formal power series, Internat. J. Control, 81(3) (2008), 336-341.
  • M. Rigo, Advanced Graph Theory and Combinatorics, with a foreword by Vincent Blondel, Computer Engineering Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2016.
  • A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Texts and Monographs in Computer Science, Springer-Verlag, New York-Heidelberg, 1978.
  • K. Schrempf, A standard form in (some) free fields: How to construct minimal linear representations, ArXiv e-prints, March 2018.
  • K. Schrempf, Linearizing the word problem in (some) free fields, Internat. J.Algebra Comput., 28(7) (2018), 1209-1230.
  • K. Schrempf, On the factorization of non-commutative polynomials (in free associative algebras), J. Symbolic Comput., 94 (2019), 126-148.
  • D. Smertnig, Factorizations of elements in noncommutative rings: a survey, in Multiplicative ideal theory and factorization theory, Springer Proc. Math. Stat., Springer, Cham, 170 (2016), 353-402.
  • R. P. Stanley, Enumerative Combinatorics, Volume 1, Second edition, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012.

A FACTORIZATION THEORY FOR SOME FREE FIELDS

Year 2020, , 9 - 42, 14.07.2020
https://doi.org/10.24330/ieja.768114

Abstract

Although in general there is no meaningful
concept of factorization in fields,
that in free associative algebras
(over a commutative field)
can be extended to their respective free field
(universal field of fractions) on the level of minimal linear representations
.
We establish a factorization theory by
providing an alternative definition of
left (and right) divisibility based
on the rank of an element and
show that it coincides with the "
classical''
left (and right) divisibility for non-commutative
polynomials.
Additionally we present an approach to factorize elements,
in particular rational formal power series,
into their (generalized) atoms.
The problem is reduced to solving a system of
polynomial equations with commuting unknowns.

References

  • N. R. Baeth and D. Smertnig, Factorization theory: from commutative to non-commutative settings, J. Algebra, 441 (2015), 475-551.
  • H. Bart, I. Gohberg, M. A. Kaashoek and A. C. M. Ran, Factorization of Matrix and Operator Functions: the State Space Method, Operator Theory: Advances and Applications, 178, Linear Operators and Linear Systems, Birkhauser Verlag, Basel, 2008.
  • J. Berstel and C. Reutenauer, Noncommutative Rational Series with Applications, Encyclopedia of Mathematics and its Applications, 137, Cambridge University Press, Cambridge, 2011.
  • P. M. Cohn, Noncommutative unique factorization domains, Trans. Amer. Math. Soc., 109 (1963), 313-331.
  • P. M. Cohn, Skew Fields, Theory of General Division Rings, Encyclopedia of Mathematics and its Applications, 57, Cambridge University Press, Cambridge, 1995.
  • P. M. Cohn, Basic Algebra, Groups, Rings and Fields, Springer-Verlag London, Ltd., London, 2003.
  • P. M. Cohn, Further Algebra and Applications, Springer-Verlag London, Ltd., London, 2003.
  • P. M. Cohn, Free Ideal Rings and Localization in General Rings, New Mathematical Monographs, 3, Cambridge University Press, Cambridge, 2006.
  • P. M. Cohn and C. Reutenauer, A normal form in free fields, Canad. J. Math., 46(3) (1994), 517-531.
  • P. M. Cohn and C. Reutenauer, On the construction of the free eld, Dedicated to the memory of Marcel-Paul Schutzenberger, Internat. J. Algebra Comput., 9(3-4) (1999), 307-323.
  • D. A. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Fourth edition, Undergraduate Texts in Mathematics, Springer, Cham, 2015.
  • FriCAS Computer Algebra System, W. Hebisch, 2018. http://axiom-wiki.newsynthesis.org/FrontPage.
  • J. W. Helton, I. Klep and J. Volcic, Geometry of free loci and factorization of noncommutative polynomials, Adv. Math., 331 (2018), 589-626.
  • B. Janko, Factorization of Non-commutative Polynomials and Testing Fullness of Matrices, Diplomarbeit, TU Graz, 2018.
  • D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting, Linear Algebra Appl., 430(4) (2009), 869-889.
  • O. Ore, Linear equations in non-commutative fields, Ann. of Math., 32(3) (1931), 463-477.
  • C. Reutenauer, Michel Fliess and non-commutative formal power series, Internat. J. Control, 81(3) (2008), 336-341.
  • M. Rigo, Advanced Graph Theory and Combinatorics, with a foreword by Vincent Blondel, Computer Engineering Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2016.
  • A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Texts and Monographs in Computer Science, Springer-Verlag, New York-Heidelberg, 1978.
  • K. Schrempf, A standard form in (some) free fields: How to construct minimal linear representations, ArXiv e-prints, March 2018.
  • K. Schrempf, Linearizing the word problem in (some) free fields, Internat. J.Algebra Comput., 28(7) (2018), 1209-1230.
  • K. Schrempf, On the factorization of non-commutative polynomials (in free associative algebras), J. Symbolic Comput., 94 (2019), 126-148.
  • D. Smertnig, Factorizations of elements in noncommutative rings: a survey, in Multiplicative ideal theory and factorization theory, Springer Proc. Math. Stat., Springer, Cham, 170 (2016), 353-402.
  • R. P. Stanley, Enumerative Combinatorics, Volume 1, Second edition, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Konrad Schrempf This is me

Publication Date July 14, 2020
Published in Issue Year 2020

Cite

APA Schrempf, K. (2020). A FACTORIZATION THEORY FOR SOME FREE FIELDS. International Electronic Journal of Algebra, 28(28), 9-42. https://doi.org/10.24330/ieja.768114
AMA Schrempf K. A FACTORIZATION THEORY FOR SOME FREE FIELDS. IEJA. July 2020;28(28):9-42. doi:10.24330/ieja.768114
Chicago Schrempf, Konrad. “A FACTORIZATION THEORY FOR SOME FREE FIELDS”. International Electronic Journal of Algebra 28, no. 28 (July 2020): 9-42. https://doi.org/10.24330/ieja.768114.
EndNote Schrempf K (July 1, 2020) A FACTORIZATION THEORY FOR SOME FREE FIELDS. International Electronic Journal of Algebra 28 28 9–42.
IEEE K. Schrempf, “A FACTORIZATION THEORY FOR SOME FREE FIELDS”, IEJA, vol. 28, no. 28, pp. 9–42, 2020, doi: 10.24330/ieja.768114.
ISNAD Schrempf, Konrad. “A FACTORIZATION THEORY FOR SOME FREE FIELDS”. International Electronic Journal of Algebra 28/28 (July 2020), 9-42. https://doi.org/10.24330/ieja.768114.
JAMA Schrempf K. A FACTORIZATION THEORY FOR SOME FREE FIELDS. IEJA. 2020;28:9–42.
MLA Schrempf, Konrad. “A FACTORIZATION THEORY FOR SOME FREE FIELDS”. International Electronic Journal of Algebra, vol. 28, no. 28, 2020, pp. 9-42, doi:10.24330/ieja.768114.
Vancouver Schrempf K. A FACTORIZATION THEORY FOR SOME FREE FIELDS. IEJA. 2020;28(28):9-42.