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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]
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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

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