Imbedding of Mal'tsev algebras into alternative algebras
Let \(\Phi\) be an associative commutative ring with identity containing 1/6. In [Algebra Logika 18, 599-613 (1979; Zbl 0442.17008)] the author has constructed nonzero nilpotent fully characteristic ideals \(G(A_ n)\) and \(K(A_ n)\) of a free Mal'cev \(\Phi\)-algebra \(A_ n\) on \(n\geq 5\) free generators. In the present paper he introduces a new function f and proves that if \(F(A_ n)\) is a fully characteristic ideal of the algebra \(A_ n\), generated by the function f, then \(G(A_ n)\subseteq F(A_ n)\subseteq K(A_ n)\). If 1/5\(\in \Phi\) and \(n\geq 6\), then \(G(A_ n)\neq F(A_ n)\); if \(n\leq 5\), then \(G(A_ n)=F(A_ n).\) Let \({\mathfrak F}\) be a subvariety of the variety of Mal'cev algebras which is defined by the identity \(f=0\) and \(A\in {\mathfrak F}\). Then the alternative enveloping algebra B(J(A)) of the ideal J(A) of the algebra A generated by its Jacobians is constructed; if the initial algebra A is finite-dimensional, then the algebra B(J(A)) is also finite-dimensional. In [Algebra Logika 19, 480-499 (1980; Zbl 0468.17006)] the author introduced the notion of a subvariety \({\mathfrak H}\) of the variety of Mal'cev algebras. In the paper under review he constructs the alternative enveloping algebra \(B(A^ 2)\) of the ideal \(A^ 2\) of an arbitrary algebra \(A\in {\mathfrak H}\). Here, from the finite-dimensionality of the algebra A follows the finite-dimensionality of the algebra \(B(A^ 2)\).
scientific article; zbMATH DE number 3855346