Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T22:55:52.209Z Has data issue: false hasContentIssue false
  • English
  • Français

HOW A NONASSOCIATIVE ALGEBRA REFLECTS THE PROPERTIES OF A SKEW POLYNOMIAL

Part of: General nonassociative rings Rings and algebras arising under various constructions

Published online by Cambridge University Press:  26 November 2019

C. BROWN  and
S. PUMPLÜN Open the ORCID record for S. PUMPLÜN [Opens in a new window]
C. BROWN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom e-mails: [email protected], [email protected]
S. PUMPLÜN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom e-mails: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let D be a unital associative division ring and D[t, σ, δ] be a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation. For each f ϵ D[t, σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f. These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f. In the classical literature, this nucleus appears as the eigenspace of f and is used to investigate the irreducible factors of f. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras Sf.

MSC classification

Primary: 16S36: Ordinary and skew polynomial rings and semigroup rings
Secondary: 17A60: Structure theory 17A99: None of the above, but in this section
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 6 - 26
Copyright
© Glasgow Mathematical Journal Trust 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Amitsur, A. S., Non-commutative cyclic fields, Duke Math. J. 21 (1954), 87–105.Google Scholar
Bergen, J., Giesbrecht, M., Shivakumar, P. N. and Zhang, Y., Factorizations for difference operators, Adv. Diff. Equ. 57 (2015), doi:10.1186/s13662-015-0402-1.Google Scholar
Boubaki, N., Éléments de mathématique. Fasc. XXIII. Algèbre; Chapitre 8, modules e anneaux semi-simples. Hermann (1973).Google Scholar
Brown, C., Petit algebras and their automorphisms. PhD Thesis (University of Nottingham, 2018). arXiv:1806.00822 [math.RA]Google Scholar
Brown, C. and Pumplün, S., The automorphisms of Petit’s algebras, Comm. Algebra 46(2) (2018), 834–849.CrossRefGoogle Scholar
Brown, C. and Pumplün, S., Nonassociative cyclic extensions of fields and central simple algebras, J. Pure Applied Algebra 223(6) (2019), 2401–2412.CrossRefGoogle Scholar
Bruck, R. H., Contributions to the theory of loops, Trans. AMS 60(2) (1946), 245–345.CrossRefGoogle Scholar
Carcanague, J., Ideaux bilaterales d’un anneau de polynomes non commutatifs sur un corps, J. Algebra 18(1) (1971), 1–18.Google Scholar
Carcanague, J., Quelques resultats sur les anneaux de Ore. C. R. Acad. Sci. Paris Sr. A-B 269 (1969), A749–A752.Google Scholar
Caruso, X. and Le Borgne, J., A new faster algorithm for factoring skew polynomials over finite fields. J. Symb. Comp. 79 (2017), 411–443.CrossRefGoogle Scholar
Churchill, R. C. and Zhang, Y., Irreducibility criteria for skew polynomials. J. Algebra 322 (2009), 3797–3822.CrossRefGoogle Scholar
Cohn, P. M., Skew fields. Theory of general division rings. Encyclopedia of Mathematics and its Applications. (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Giesbrecht, M., Factoring in skew-polynomial rings over finite fields, J. Symbolic Comput. 26(4) (1998), 463–486.CrossRefGoogle Scholar
Giesbrecht, M. and Zhang, Y., Factoring and decomposing Ore polynomials over $\mathbb{F}_q(t)$ , Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ACM, New York, 2003), 127–134.Google Scholar
Gòmez-Torrecillas, J., Basic module theory over non-commutative rings with computational aspects of operator algebras. With an appendix by V. Levandovskyy, in, Algebraic and algorithmic aspects of differential and integral operators, LNCS, Vol. 8372 (Springer, Heidelberg, 2014), 23–82.Google Scholar
Goodearl, K., Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150(2) (1992), 324–377.CrossRefGoogle Scholar
Gòmez-Torrecillas, J., Lobillo, F. J. and Navarro, G., Computing the bound of an Ore polynomial. Applications to factorization. J. Symbolic Comp. (2018), https://doi.org/10.1016/j.jsc.2018.04.018.Google Scholar
Granja, A., Martinez, C. and Rodriguez, C., Real Valuations on Skew Polynomial Rings, Algebr. Represent. Th. 17(5) (2014), 1413–1436.CrossRefGoogle Scholar
Hungerford, T. W., Algebra. Vol. 73. Graduate Texts in Mathematics (Springer, New York, 1980).CrossRefGoogle Scholar
Jacobson, N., Finite-Dimensional Division Algebras Over Fields (Springer, Berlin-Heidelberg-New York, 1996).Google Scholar
Jacobson, N., Pseudo-linear transformations. Ann. Math. 38(2) (1937), 484–507.CrossRefGoogle Scholar
Jacobson, N., The Theory of Rings (AMS, Providence, RI, 1943).Google Scholar
Koblitz, N., A course in number theory and cryptography. Vol. 114 (Springer Science and Business Media, 1994).Google Scholar
Lavrauw, M. and Sheekey, J., Semifields from skew-polynomial rings, Adv. Geom. 13(4) (2013), 583–604.CrossRefGoogle Scholar
Lam, T. Y. and Leroy, A., Vandermonde and Wroskian matrices over division rings, J. Alg. 19(2)(1988), 308–336.CrossRefGoogle Scholar
Lam, T. Y. and Leroy, A., Algebraic conjugacy classes and skew-polynomial rings, in Perspectives in Ring Theory (Springer, Antwerp, 1988), 153–203.Google Scholar
Lam, T. Y., Leung, K. H., Leroy, A. and Matczuk, J., Invariant and semi-invariant polynomials in skew polynomial rings, in Ring theory 1989 (Ramat Gan and Jerusalem, 1988/1989), 247–261, Israel Math. Conf. Proc., Vol. 1 (Weizmann, Jerusalem, 1989).Google Scholar
Lemonnier, B., Dimension de Krull et codeviations, quelques applications en theorie des modules. PhD Thesis, Poitiers (1984).Google Scholar
Ore, O., Theory of noncommutative polynomials, Annals of Math. 34(3) (1933), 480–508.CrossRefGoogle Scholar
Petit, J.-C., Sur certains quasi-corps généralisant un type d’anneau-quotient, Séminaire Dubriel. Algèbre et théorie des nombres 20 (1966–67), 1–18.Google Scholar
Petit, J.-C., Sur les quasi-corps distributifes à base momogène. C. R. Acad. Sc. Paris 266 (1968), Série A, 402–404.Google Scholar
Pumplün, S., Factoring skew polynomials over Hamilton’s quaternion algebra and the complex numbers, J. Algebra 427 (2015), 20–29.Google Scholar
Pumplün, S., Corrigendum to: Factoring skew polynomials over Hamilton’s quaternion algebra and the complex numbers, J. Algebra 427 (2015), 20–29. http://agt2.cie.uma.es/~loos/jordan/ CrossRefGoogle Scholar
Pumplün, S., Nonassociative differential extensions of characteristic p. Results in Mathematics 72(1–2) (2017), 245–262.Google Scholar
Pumplün, S., Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes, Adv. Math. Comm. 11(3) (2017), 615–634. doi: 10.3934/amc.2017046.Google Scholar
Pumplün, S., Tensor products of nonassociative cyclic algebras, J. Algebr. 451 (2016), 145–165.CrossRefGoogle Scholar
Pumplün, S., Algebras whose right nucleus is a central simple algebra. J. Pure and Applied Algebra 222(9) (2018), 2773–2783. https://doi.org/10.1016/j.jpaa.2017.10.019.Google Scholar
Pumplün, S., How to obtain lattices from (f, σ, δ)-codes via a generalization of construction A, Appl. Algebra Engrg. Comm. Comput. 29(4) (2018), 313–333.Google Scholar
Pumplün, S., Quotients of orders in algebras obtained from skew polynomials with applications to coding theory, Comm. Algebra 46(11) (2018), 5053–5072.CrossRefGoogle Scholar
Rónyai, L., Factoring polynomials over finite fields, J. Algorithms 9(3) (1988), 391–400.Google Scholar
Schafer, R. D., An Introduction to Nonassociative Algebras (Dover Publ., Inc., New York, 1995).Google Scholar