Alternative Rings and Related Questions I: Existence of the Radical

Abstract

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The Wedderburn principal theorem for alternative algebras

“…1 See references [6], [7], [9]. Numbers in brackets refer to the references cited at the end of the paper.…”

“…Furthermore, if 21 is separable, there exists a scalar extension $ of finite degree over § such that 21$ is a direct sum of components each of which is either a total matric algebra or a Cayley-Dickson algebra with divisors of zero. 6 Such a scalar extension Ü of 8 we call a splitting field of 21, and we use the term split algebra for a total matric algebra or a Cayley-Dickson algebra with divisors of zero. The number of total matric components of 21$ is the same for all splitting fields $ of 21, and is the sum of the degrees over S of the centers of the associative simple components of 21.…”

“…It follows from an observation of Etherington [4, p. 2] that 5ft is nilpotent if and only if the associative algebra 5ft* generated by the right, and left multiplications of Sft is nilpotent (for a simple induction suffices to prove that, if yi* k = 0, then every product of / = 2* elements of 9t is zero). It is an immediate consequence of the known structure theory for alternative algebras and of the following lemma that the radical 91 of an alternative algebra 21 is the 6 [6, equation (5) 9 The term "strongly nilpotent 7 ' has been used for this concept recently, and "nilpotent" used in case the associative algebra $1* were nilpotent. However, Etherington's theorem shows the equivalence of the two notions.…”

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“…We have shown this for the Cayley components [10, Theorem 6]. For the associative components the conclusion follows from (25) and (12). See also Theorem 5…”

Alternative regular rings without nilpotent elements

“…Preliminaries. Max Zorn calls a ring R alternative in case, for every a, b, cÇzR, the associator (a, b, c) ~a(bc) -(ab)c changes sign on interchange of two of its arguments [7,8]. Interest in these rings currently stems from a fundamental result of R. Moufang in the foundations of projective geometry [5].…”