Mal’tsevh 0-algebras

Abstract

Let Φ be an associative commutative ring with unity, 1/6 ∈ Φ, write A for a Mal’tsev algebra over Φ, suppose that on A, the function h(y, z, t, x, x)=2[{yz, t, x}x+{yx, z, x}t], where {x, y, z}=(xy)z−(xz)y+2x(yz), is defined, and assume that H(A) is a fully invariant ideal of A generated by the function h. The algebra A satisfying an identity h(y, z, x, x, x)=0 [h(y, z, t, x, x)=0] is called a Mal’tsev h0-algebra (h-algebra). We prove that in any Mal’tsev h0-algebra, the inclusion H(A)·A2⊆Ann A holds withAnnA the annihilator of A. This means that any semiprime h0-algebra A is an h-algebra. Every prime h0-algebra A is a central simple algebra over the quotient field Λ of the center of its algebra of right multiplications, R(A), and is either a 7-dimensional non-Lie algebra or a 3-dimensional Lie algebra over Λ.