JOURNAL OF ALGEBRA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF 147, 81 89 (1992) On the J-Potency of Malcev Algebras ALBERT-• EI.DUQUE* zyxwvutsrqponmlkjihgfedcbaZYX Depurtumento de Matemtiticus, Unicersidad de Zaragox. XM W YZuragoza. Spain Communicuted by E. Kleinfeld Received March 7. 1990 I. INTRODUCTION A Malcev algebra is a nonassociative algebra satisfying the identities .$ = 0 (1) J( X, )‘, X2) = J( .Y, ): z).X-, where J(.u, _Y, -_) = (.Y_Y)z + (yz).r + (zx) y is the Jacobian of the elements x, y, and L’. Throughout the paper we will deal with linite dimensional Malcev algebras defined over a field F of characteristic not two. These two hypotheses will be tacitly assumed. If M is a Malcev algebra, the subsets J(M, M, M), spanned by the Jacobians, and N(M) = { .Y E M zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK : J(.r , M, M) = 0 }, called the J-nucleus of M, are ideals of the algebra which help us in measuring the “Lie-ness” of M. It is clear that a Malcev algebra is a Lie algebra if and only if J( M, M, M) = 0 or, equivalently, N(M) = M. Now, let B be an ideal of the Malcev algebra M. we deline the subspaces - J;(t)=~(B, B, B), JI;+‘(B)=J(J;j’(B),J;;‘(B),J;;(B)). - J;(B)=J(B, B, M), J~ +‘(B)=J(J~ (B),J’,“(B), M). - J;(B)=J(B, M, M), JF+‘=J(J~(B), M, M). It should be noted that Jy( B) is an idea1 of M [ 11, Theorem 3.53 for i = 1, 2. The ideal B is called J,-potent if there exists an n such that J:(B) = 0 (i = 0, 1, 2). This definition is due to Sagle [ 111, although he assumed n to be greater than one. But, as in [lo] rr’e ~ilf not do so. Obviously Jz( B) z Jy( B) G Jy( B) for every m. * Partially supported by the DGICYT (PS 87-0054). 81 002 I -8693,‘92 $3.00 (‘op)r!ght I 1992 b) Acadcmlc Prcrs. Inc All right, ItI rcprnducllon I” anykm rcscrvcd.