Previous |  Up |  Next


Skew polynomial ring; reducible skew polynomials; eigenspace; nonassociative algebra; semisimple Artinian ring.
We find examples of polynomials $f\in D[t;\sigma ,\delta ]$ whose eigenring $\mathcal {E}(f)$ is a central simple algebra over the field $F = C \cap \mathrm {Fix}(\sigma ) \cap \mathrm {Const}(\delta )$.
[1] Albert, A.A.: On nonassociative division algebras. Transactions of the American Mathematical Society, 72, 2, 1952, 296-309, JSTOR, DOI 10.1090/S0002-9947-1952-0047027-4
[2] Amitsur, A.S.: Differential polynomials and division algebras. Annals of Mathematics, 1954, 245-278, JSTOR, DOI 10.2307/1969691
[3] Amitsur, A.S.: Non-commutative cyclic fields. Duke Mathematical Journal, 21, 1, 1954, 87-105, Duke University Press, DOI 10.1215/S0012-7094-54-02111-0
[4] Amitsur, A.S.: Generic splitting fields of central simple algebras. Annals of Mathematics, 62, 2, 1955, 8-43, JSTOR, DOI 10.2307/2007098
[5] Bourbaki, N.: Elements of mathematics. 2003, Springer,
[6] Brown, C., Pumplün, S.: How a nonassociative algebra reflects the properties of a skew polynomial. Glasgow Mathematical Journal, 63, 1, 2021, 6-26, Cambridge University Press, DOI 10.1017/S0017089519000478
[7] Brown, C.: Petit algebras and their automorphisms. 2018, PhD Thesis, University of Nottingham. Online at arXiv:1806.00822.
[8] Carcanague, J.: Quelques résultats sur les anneaux de Ore. CR Acad. Sci. Paris Sr. AB, 269, 1969, A749-A752,
[9] Cohn, P.M.: Noncommutative unique factorization domains. Transactions of the American Mathematical Society, 109, 2, 1963, 313-331, DOI 10.1090/S0002-9947-1963-0155851-X
[10] Gòmez-Torrecillas, J., Lobillo, F. J., Navarro, G.: Computing the bound of an Ore polynomial. Applications to factorization. Journal of Symbolic Computation, 92, 2019, 269-297, Elsevier, DOI 10.1016/j.jsc.2018.04.018
[11] Gòmez-Torrecillas, J.: Basic module theory over non-commutative rings with computational aspects of operator algebras. With an appendix by V. Levandovskyy. Algebraic and Algorithmic Aspects of Differential and Integral Operators. AADIOS 2012. Lecture Notes in Computer Science, 8372, 2014, 23-82, Springer,
[12] Goodearl, K. R., Bruce, J. W., Warfield, R. B.: An introduction to noncommutative Noetherian rings. 61, 2004, Cambridge University Press,
[13] Jacobson, N.: Finite-dimensional division algebras over fields. 1996, Springer,
[14] Jacobson, N.: The theory of rings. 2, 1943, American Mathematical Society,
[15] McConnell, J.C., Robson, C.J., Small, L.W.: Noncommutative noetherian rings. 30, 2001, American Mathematical Soc.,
[16] Ore, O.: Theory of non-commutative polynomials. Annals of Mathematics, 1933, 480-508, JSTOR, DOI 10.2307/1968173 | Zbl 0007.15101
[17] Owen, A.: On the right nucleus of Petit algebras. PhD Thesis, University of Nottingham, in preparation..
[18] Pumplün, S.: Algebras whose right nucleus is a central simple algebra. Journal of Pure and Applied Algebra, 222, 9, 2018, 2773-2783, Elsevier, DOI 10.1016/j.jpaa.2017.10.019
Partner of
EuDML logo