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By Richard D. Schafer

ISBN-10: 0080873340

ISBN-13: 9780080873343

ISBN-10: 0123745691

ISBN-13: 9780123745699

ISBN-10: 0126224501

ISBN-13: 9780126224504

An creation to Nonassociative Algebras Richard D. Schafer

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An creation to Nonassociative Algebras Richard D. Schafer

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Suppose that s,,, sij are in G i j ; then sijajiand sljaj, in a,, belong to the ideal of nilpotent elements of aii,and (asij + /ls;i)ajiis nilpotent for all a, p in F. That is, asfj psij is in G,,, and G,j is a subspace of '$I. , t) is properly nilpotent. 7 (Zorn). The radical% (= maximalnilideal) of anyfinitedimensional alternative algebra 2l is the set Cp of all properly nilpotent elements of a. Also denote by $3 the subspace of spanned by Cp. 6. We shall show that G = $3 is an ideal of a. $3, x $6 = 6,.

Then dim But u E 2111,e,,u $ c dim 2111,e,, and this process of increasing dimensions must terminate, yielding a principal idempotent. We note that, if u is any idempotent in a finite-dimensional alternative algebra %, then there exist pairwise orthogonal primitive idempotents el, ... , e,, ... , e, in % (1 I r < t ) such that u = e l + +e, while e = el + ... + e, is a principal idempotent in %. ,e , of pairwise orthogonal primitive idempotents e i such that e = el + ... + e, is a principal idempotent of %.

K). 35) implies C1 = e z , Zz = el. Hence b=Be,- Caiui- CBjwj+aez. 45) that + C aiui g12 + 1Bj w j h 1 + B h 1 = B h z l - 1atg1, ui - C P j h 1 wj + agz1 bu = agiz = ub. 40) holds for all b in 23. 38) with p = 1, and b + ub is a homomorphic image in 2l of the (2n)-dimensional algebra constructed by the Cayley-Dickson process from the algebra 23 of dimension n = 2 + 2k. Beginning with the 2-dimensional algebra Fe, @ Fez , the process does not terminate until we have 23 = a. By (iii) a # Fe, @ Fez.

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An Introduction to Nonassociative Algebras by Richard D. Schafer


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