Annihilators of Mal'tsev algebras
Let \(\Phi\) be an associative ring with 1, containing \({1\over 6}\), and let \(A\) be a Malcev \(\Phi\)-algebra. Let \[ \{x, y, z\}= (xy)z- (xz)y+ 2x(yz), \] \[ J(x, y, z)= (xy)z+ (zx)y+ (yz)x, \] \[ h(y, z, t, u, x)= u{\textstyle {\partial \over {\partial x}}} \bigl[\{yz, t, x\}x+ \{yx, z, x\}t \bigr]\qquad \text{ and} \] \[ g(y, z, t, v, u, x)= u{\textstyle {\partial\over {\partial x}}} \bigl[J(\{yz, t, x\}, x,v)+ J(\{yx, z, x\}, t, v) \bigr], \] where \(u (\partial/\partial x)\) is the operator of differential substitution \(x\to u\). Let \(G(A) (H(A))\) be the fully invariant ideal of \(A\) generated by \(g(h)\). In the present paper, it is shown that \(G(A) \cdot A^2 \subseteq \text{Ann} (A)\) and that \(H(A) J(A) \subseteq \text{Ann} (A)\). A consequence is that an algebra of characteristic 0 which satisfies the \(n\)-th Engel condition has \(A^n \cdot A^2 \subseteq \text{Ann} (A)\) for some \(n\). These are but a sample of the many such results in this paper.
scientific article; zbMATH DE number 848945