Automorphically-invariant ideals satisfying multilinear identities, and group-theoretic applications

Abstract

Let A be an arbitrary (not necessarily associative or commutative) algebra over a field K. It is proved that if A has an ideal of finite codimension r satisfying a multilinear identity f ≡ 0 , then A also has an ideal satisfying the same identity f ≡ 0 that is invariant under all automorphisms of A and has finite codimension bounded in terms of r and f. The result is stronger in characteristic zero, where f need not be multilinear. As a corollary, it is proved that if a locally nilpotent torsion-free group G has a normal subgroup H satisfying a multilinear commutator identity ϰ ( H ) ≡ 1 with quotient G / H of finite rank r, then G also has a characteristic subgroup C satisfying the same identity ϰ ( C ) ≡ 1 with quotient G / C of finite rank bounded in terms of r and ϰ. An example shows that the main result cannot be extended to algebras not over fields, even to Lie algebras over integers. An analogous example shows that the result on characteristic nilpotent subgroups with quotients of finite rank, which was proved by the authors earlier in torsion-free and periodic cases, cannot be extended to mixed nilpotent groups.