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A nonassociative ring is called a derivation alternator ring if it satisfies the identities ( y z , x , x ) = y ( z , x , x ) + ( y , x , x ) z , ( x , x , y z ) = y ( x , x , z ) + ( x , x , y ) z (yz,\,x,\,x)\, = \,y(z,\,x,\,x)\, + \,(y,\,x,\,x)z,\,(x,\,x,\,yz)\, = \,y(x,\,x,\,z)\, + \,(x,\,x,\,y)z and ( x , x , x ) = 0 (x,\,x,\,x)\, = 0 . Let R be a prime derivation alternator ring with idempotent e ≠ 1 e \ne 1 and characteristic ≠ 2 \ne 2 . If R is without nonzero nil ideals of index 2, then R is alternative.

We show that in any right alternative algebra, the additive span of the alternators is nearly an ideal. We give an easy test to use to determine if a given set of additional identities will imply that the span of the alternators is an ideal. We apply our technique to the class of right alternative algebras satisfying the condition ( a , a , b ) = λ [ a , [ a , b ] ] (a,a,b) = \lambda [a,[a,b]] . We show that any semiprime algebra over a field of characteristic ≠ 2 \ne 2 , ≠ 3 \ne 3 which satisfies the right alternative law and the above identity with λ ≠ 0 \lambda \ne 0 is a subdirect sum of (associative and commutative) integral domains.

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).

If I is the ideal generated by all associators, ( a , b , c ) = ( a b ) c − a ( b c ) (a,b,c) = (ab)c - a(bc) , it is well known that in any nonassociative algebra R , I ⊆ ( R , R , R ) + R ( R , R , R ) R,I \subseteq (R,R,R) + R(R,R,R) . We examine nonassociative algebras where I ⊆ ( R , R , R ) I \subseteq (R,R,R) . Such algebras include ( − 1 , 1 ) ( - 1,1) algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), (a, b, a), (b, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where I = ( R , R , R ) I = (R,R,R) as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form [ x , ( x , x , a ) ] = γ ( x , x , [ x , a ] ) [x,(x,x,a)] = \gamma (x,x,[x,a]) for fixed γ \gamma . With two exceptions, if this algebra has an idempotent e such that ( e , e , R ) = 0 (e,e,R) = 0 , then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.