A fully invariant ideal of alternative algebras

Lei φ be an associative commutative ring with I, containing 1/6, and A be an alternative φalgebra. Let D be an associator ideal of A and H a fully invariant ideal of A, generated by all elements oftheformh(y,z,t,x,x) = [{[y,z],t,x}-, x] + [{[y,x},z,x}-,t], where [x,y] = xy-yx, {x,y, z}- = [[z, y],z] - [[i,z],y] + 2[z,[y, z]]. Here we consider an ideal Q = H ∩ D and prove that Q4 = 0 in the algebra A. If A is unmixed, then HD = 0, DE = 0, and Q2 = 0 in particular. If A is a finitely generated unmixed algebra, then the ideal H lies in its associative center and Q = 0. It follows that any finitely generated purely alternative algebra satisfies the identity h(y,z,t,x,x) = 0. We also show that a fully invariant ideal HO of the unmixed algebra A, generated by all elements of the form h(x,z,t,x,x), lies in its associative center and H0∩D =0. Consequently, every purely alternative algebra satisfies the identity h(x,z,t,x,x) = 0. © 1997 Plenum Publishing Corporation.