By Richard D. Schafer
An advent to Nonassociative Algebras Richard D. Schafer
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3. 29) by induction on m. k k giving the case m = 1. 29). 20). 31) are left as exercises for the reader. 3. Note that z is properly nilpotent if and only if az is nilpotent for all a in '%. 1. Also z properly nilpotent implies z is nilpotent (since z 2 is). We shall show that the radical (= maximal nilideal) 3 of any finitedimensional alternative algebra '% is the set '$ of all properly nilpotent elements of PI. CIearly 3 5 $'3, since z in 3 implies za is in 3 for all a in '? In the associative case, the proof that 'p c 3 requires only a few lines (Albert , p.
10) R,’ = R,] , L,’= L,, for all x in a. An element x in a power-associative algebra 2l is called nilpotent in case there is an integer r such that x‘ = 0. An algebra (ideal) consisting only of nilpotent elements is called a nilalgebra (nilideal). 2. Any alternative nilalgebra PI of jinite dimension over F is nilpotent. Proof. Let b be a subalgebra of % ’ which is maximal with respect to the property that %* is nilpotent. Such a maximal b exists, since the subalgebra ( 0 ) has this property and 2l is finite-dimensional.
74) to prove a (partial) generalization to arbitrary nonassociative algebras. 5 below we require the elementary fact that, if a field K is a finite separable extension of F, then the only derivation of K (over F) is 0 (Zariski and Samuel [l], p. 124). 5 (Jacobson). Let % be a finite-dimensional algebra which is a direct sum 2l = 6 , @ 0 6,of simpre ideals 6, over F of characteristic 0, and let % contain a left (or right) identity. Then every derivation D of 2[ is inner. Proof. It is sufficient to prove this for simple algebras 2l.