By Richard D. Schafer
Concise research provides in a quick house the various very important rules and leads to the idea of nonassociative algebras, with specific emphasis on replacement and (commutative) Jordan algebras. Written as an advent for graduate scholars and different mathematicians assembly the topic for the 1st time. "An vital addition to the mathematical literature"—Bulletin of the yank Mathematical Society.
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Additional resources for An Introduction to Nonassociative Algebras
For j = k, (19 ) is obvious; for j = k, (19 ) reduces to (15). Now the radical of D+ (consisting of nilpotent elements) is 0. Although our proof of the Corollary to Theorem 7 is valid only for characteristic 0, we remarked in IV that the conclusion is valid for characteristic = 0. Hence D+ is a direct sum S1 ⊕ · · · ⊕ Sr of r simple ideals Si , each with unity element ei . The existence of an idempotent e = 1 in D+ is sufficient to give zero divisors in D, a contradiction, since the product e(1 − e) = 0 in D.
Then by Witt’s theorem (Jacobson, ibid, p. 162; Artin, ibid, p. 121), since n(x) and n (x ) are equivalent, the restrictions of n(x) and n (x ) to B⊥ and ALTERNATIVE ALGEBRAS 27 B ⊥ are equivalent. Choose v in B⊥ with n(v) = 0; correspondingly we have v in B ⊥ such that n (v ) = n(v). Then a + vb → aH0 + v (bH0 ) is an isomorphism of B ⊥ vB onto B ⊥ v B by the construction above. Hence if we begin with B = F 1, B = F 1 , repetition of the process gives successively isomorphisms between Z and Z , Q and Q , C and C .
If A is a commutative power-associative algebra over F of characteristic = 2, 3, 5, then A is strictly power-associative. The assumption of strict power-associativity is employed in the noncommutative case, and in the commutative case of characteristic 3 or 5, when one wishes to use the method of extension of the base field. Let A be a finite-dimensional power-associative algebra over F . Just as in the proofs of Propositions 1 and 2, one may argue that A has a unique maximal nilideal N, and that A/N has maximal nilideal 0.
An Introduction to Nonassociative Algebras by Richard D. Schafer