On a symmetric invariant bilinear form of the \(J\)-ideal of a semiprime Mal'tsev algebra (Q1317628)

Let \(\Phi\) be a commutative-associative ring with 1/6, \(A\) be a Mal'tsev \(\Phi\)-algebra, and \(J(A)\) be the ideal generated by all Jacobians in \(A\). (1) If \(a \in J(A)\) is an absolute zero divisor, then the ideal generated by \(a\) is solvable of index \(\leq 5\). (2) Let \(Z\) be the center of the restriction to \(J(A)\) of the right multiplications algebra \(R(A)\). The author constructs a symmetric invariant bilinear \(Z\)-form \((x,y)\) such that the identity \[ 3(x,y) z-3 (x,z)y=(xz) y-(xy)z+2x(zy) \] holds for \(x,y,z \in J(A)\). Conversely, if \(J(A)\) is semiprime, then a symmetric invariant bilinear \(Z\)-form \((x,y)\) defined on \(J(A)\) and satisfying this indicated identity is nondegenerate and must be the form constructed here. (In particular, if \(A\) is semiprime, then \(I\cap J(A)\) is semiprime for \(I\) any ideal of \(A\); and so \(J(A)\) is semiprime.) (3) These results are then used to derive some new identities. This includes one of degree 9 which holds in \(A\) if \(J(A)\) is semiprime, but does not hold in general.