Generalization of right alternative rings

Abstract

We study nonassociative rings R R satisfying the conditions (1) ( a b , c , d ) + ( a , b , [ c , d ] ) = a ( b , c , d ) + ( a , c , d ) b (ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b for all a , b , c , d ∈ R a,b,c,d \in R , and (2) ( x , x , x ) = 0 (x,x,x) = 0 for all x ∈ R x \in R . We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent ≠ 0 , ≠ 1 \ne 0, \ne 1 , are right alternative rings. We show by example that ( x , e , e ) (x,e,e) may be nonzero. In general, R = R ′ + ( R , e , e ) R = R’ + (R,e,e) (additive direct sum) where R ′ R’ is a subring and ( R , e , e ) (R,e,e) is a nilpotent ideal which commutes elementwise with R R . We examine R ′ R’ under the added assumption of Lie admissibility: (3) ( a , b , c ) + ( b , c , a ) + ( c , a , b ) = 0 (a,b,c) + (b,c,a) + (c,a,b) = 0 for all a , b , c ∈ R a,b,c \in R . We generate the Peirce decomposition. If R ′ R’ has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of R ′ R’ contains R 10 ′ + R 01 ′ {R’_{10}} + {R’_{01}} . Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3).

Generalized right alternative rings

“…Given (1), then (2) holds <=> the ring is powerassociative. Given (1) and (2), then (3) holds t=> the ring under the symmetric product a°b = ab + ba is a Jordan ring. To simplify the notation, dot and juxtaposition will be used to indicate multiplication.…”

“…Property (4) is a linearization of property (2). To show (5), it will suffice to show 0 = [a, (b,a,a)].…”

“…Hypotheses similar to (1), (2), and (3) have been studied. In [6] and [3], the three identities listed below were assumed.…”

“…In [6] and [3], the three identities listed below were assumed. (1) and (2). They have shown that when such rings are simple and possess an idempotent, then they must be right alternative.…”