By Richard D. Schafer
An creation to Nonassociative Algebras Richard D. Schafer
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E, in a. An element e of an (arbitrary) algebra 2l is called an idempotent in case ez = e z 0. 3. Any finite-dimensional power-associative algebra 2l, which is not a nilalgebra, contains an idempotent e (ZO). Proof. 2l contains an element x which is not nilpotent. The subalgebra F[x]of 2l generated by xis a finite-dimensional associative algebra which is not a nilalgebra. Then F[x] contains an idempotent e (ZO)(Albert ,p. 23), and therefore does. 1) L, and Re are idempotent operators on 2l which commute by the flexible law (they are “commuting projections”).
T(x) - t(ax and If a # 0 and obtain p # 0, we may divide by these scalars, and subtract, to (a - B)[t(x) + 0)- 0 + Y)l = 0. 41). In order to facilitate the passage to scalar extensions, we wish to have a linear trace. We are led therefore to the following definition. IU. AlternativeAlgebras 50 Let 2l # F1 be an algebra with 1 over F such that for each x in % we have xz - t(x)x + n(x)l = 0, t(x), n(x) in F; in addition, if I; is the field of two elements, the trace r(x) (uniquely defined by setting t(a1) = 2a) is required to be linear.
19) and the fact that associators alternate that is associative in case alland are associative and = '%gl = 0. But III. Alternative Algebras 52 Hence = ail = 0. + e,, we have aloe associative. Hence 'illis associative. 16. I be a finite-dimensional simple alternative algebra over (an arbitrary field) F satisfying: (i) 1 = el + e2 for (orthogonal) primitive idempotents e,; (ii) %,l(=e,91e,)=Fe, (i=l,2); (iii) 2l is not associative. 38) with p = 1. Proof. Let 23 be any semisimple subalgebra of which contains el and e, .