Title  Alternative algebras and RA loops 
Author  Zhou, Yongxin, 1964 
Description  Thesis (Ph.D.)Memorial University of Newfoundland, 1999. Mathematics and Statistics 
Date  1998. 
Pagination  ix, 122 leaves ; 28 cm. 
Subject  Alternative algebras; Loops (Group theory); 
Degree  Ph.D. 
Degree Grantor  Memorial University of Newfoundland. Dept. of Mathematics and Statistics

Discipline  Mathematics and Statistics

Language  Eng 
Notes  Bibliography: leaves 119122. 
Abstract  In the first part of this thesis, we study the relationships between three algebra structures: CayleyDickson 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 (α, β, γ) be the CayleyDickson algebra. We construct an RA loop L from each CayleyDickson algebra A (α, 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 CayleyDickson algebras and the category of RA loops, we show that the two categories are equivalent.  Using the induced RA loops, we show that any CayleyDickson algebra is a homomorphic image of an alternative loop algebra. Thus we give a new way of representing a CayleyDickson algebra. Furthermore, the homomorphism commutes with the norm and trace operations of the alternative loop algebra and the CayleyDickson algebra. The kernel of this homomorphism is completely determined. The prime radical and Jacobson radical of some CayleyDickson 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 groupgraded alternative algebras is derived. 
Type  Text 
Resource Type  Electronic thesis or dissertation 
Format  Image/jpeg; Application/pdf 
Source  Paper copy kept in the Centre for Newfoundland Studies, Memorial University Libraries 
Local Identifier  a1358218 
Rights  The 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. 
Collection  Electronic Theses and Dissertations 
Scanning Status  Completed 
PDF File  (13.67 MB)  http://collections.mun.ca/PDFs/theses/Zhou_Yongxin.pdf 
CONTENTdm file name  21012.cpd 