Research Article
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Year 2020, Volume 28, Issue 28, 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, Volume 28, Issue 28, 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.

Details

Primary Language English
Subjects Mathematics
Journal Section Articles
Authors

Konrad SCHREMPF This is me (Primary Author)
University of Vienna
Austria

Publication Date July 14, 2020
Published in Issue Year 2020, Volume 28, Issue 28

Cite

Bibtex @research article { ieja768114, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2020}, volume = {28}, number = {28}, pages = {9 - 42}, doi = {10.24330/ieja.768114}, title = {A FACTORIZATION THEORY FOR SOME FREE FIELDS}, key = {cite}, author = {Schrempf, Konrad} }
APA Schrempf, K. (2020). A FACTORIZATION THEORY FOR SOME FREE FIELDS . International Electronic Journal of Algebra , 28 (28) , 9-42 . DOI: 10.24330/ieja.768114
MLA Schrempf, K. "A FACTORIZATION THEORY FOR SOME FREE FIELDS" . International Electronic Journal of Algebra 28 (2020 ): 9-42 <https://dergipark.org.tr/en/pub/ieja/issue/55997/768114>
Chicago Schrempf, K. "A FACTORIZATION THEORY FOR SOME FREE FIELDS". International Electronic Journal of Algebra 28 (2020 ): 9-42
RIS TY - JOUR T1 - A FACTORIZATION THEORY FOR SOME FREE FIELDS AU - KonradSchrempf Y1 - 2020 PY - 2020 N1 - doi: 10.24330/ieja.768114 DO - 10.24330/ieja.768114 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 9 EP - 42 VL - 28 IS - 28 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.768114 UR - https://doi.org/10.24330/ieja.768114 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra A FACTORIZATION THEORY FOR SOME FREE FIELDS %A Konrad Schrempf %T A FACTORIZATION THEORY FOR SOME FREE FIELDS %D 2020 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 28 %N 28 %R doi: 10.24330/ieja.768114 %U 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
AMA Schrempf K. A FACTORIZATION THEORY FOR SOME FREE FIELDS. IEJA. 2020; 28(28): 9-42.
Vancouver Schrempf K. A FACTORIZATION THEORY FOR SOME FREE FIELDS. International Electronic Journal of Algebra. 2020; 28(28): 9-42.
IEEE K. Schrempf , "A FACTORIZATION THEORY FOR SOME FREE FIELDS", International Electronic Journal of Algebra, vol. 28, no. 28, pp. 9-42, Jul. 2020, doi:10.24330/ieja.768114