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Memorial University - Electronic Theses and Dissertations 4
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Document Description
TitleAlternative algebras and RA loops
AuthorZhou, Yongxin, 1964-
DescriptionThesis (Ph.D.)--Memorial University of Newfoundland, 1999. Mathematics and Statistics
Paginationix, 122 leaves ; 28 cm.
SubjectAlternative algebras; Loops (Group theory);
Degree GrantorMemorial University of Newfoundland. Dept. of Mathematics and Statistics
DisciplineMathematics and Statistics
NotesBibliography: leaves 119-122.
AbstractIn the first part of this thesis, we study the relationships between three algebra structures: Cayley-Dickson algebras, RA loops and alternative loop algebras. -- Let R be a commutative associative ring with 1 and let A be an R -algebra with unity of characteristic different from 2. For any α, β and γ A , let A , β, γ) be the Cayley-Dickson algebra. We construct an RA loop L from each Cayley-Dickson algebra A , β, γ), called the induced RA loop. We show that any RA loop is a homomorphic image of some induced RA loop. After introducing the category of Cayley-Dickson algebras and the category of RA loops, we show that the two categories are equivalent. -- Using the induced RA loops, we show that any Cayley-Dickson algebra is a homomorphic image of an alternative loop algebra. Thus we give a new way of representing a Cayley-Dickson algebra. Furthermore, the homomorphism commutes with the norm and trace operations of the alternative loop algebra and the Cayley-Dickson algebra. The kernel of this homomorphism is completely determined. The prime radical and Jacobson radical of some Cayley-Dickson algebras are determined. A result of de Barros is generalized. The more general form of the homomorphism is studied. -- Necessary and sufficient conditions for an RA loop to be the Moufang circle loop of a quasiregular alternative algebra are given. The algebra structure of a finite alternative nilpotent ring with the Moufang circle loop being an RA loop is completely determined. -- In the second part of this thesis, the alternative rings of order p4 and p5 are completely determined, where p is a prime. This generalizes a result of A. T. Gainov. The two smallest alternative rings have order 24. For each prime number, there are fifteen alternative rings of order pn , n 5. The relationships between these fifteen rings are described. From these alternative algebras, a class of group-graded alternative algebras is derived.
Resource TypeElectronic thesis or dissertation
FormatImage/jpeg; Application/pdf
SourcePaper copy kept in the Centre for Newfoundland Studies, Memorial University Libraries
Local Identifiera1358218
RightsThe author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission.
CollectionElectronic Theses and Dissertations
Scanning StatusCompleted
PDF File(13.67 MB) --
CONTENTdm file name21012.cpd