How a nonassociative algebra reflects the properties of a skew polynomial

Brown, Christian; Pumpluen, Susanne

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 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.

Citation

Brown, C., & Pumpluen, S. (2021). How a nonassociative algebra reflects the properties of a skew polynomial. Glasgow Mathematical Journal, 63(1), 6-26. https://doi.org/10.1017/S0017089519000478

• Journal Article Type: Article
• Acceptance Date: Oct 25, 2019
• Online Publication Date: Nov 26, 2019
• Publication Date: 2021-01
• Deposit Date: Oct 31, 2019
• Publicly Available Date: May 27, 2020
• Journal: Glasgow Mathematical Journal
• Print ISSN: 0017-0895
• Electronic ISSN: 1469-509X
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 63
• Issue: 1
• Pages: 6-26
• DOI: https://doi.org/10.1017/S0017089519000478
• Keywords: Rings and Algebras
• Public URL: https://nottingham-repository.worktribe.com/output/2461416
• Publisher URL: https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/how-a-nonassociative-algebra-reflects-the-properties-of-a-skew-polynomial/BFA5A014A86241EB7F0536260A971DD1
• Additional Information: Once published, add set statement to Additional Information: This article has been published in a revised form in Glasgow Mathematical Journal, https://doi.org/10.1017/S0017089519000478 This version is free to view and download for private research and study only. Not for re-distribution or re-use. © copyright holder.
• Contract Date: Oct 31, 2019