Fully noncentral Lie ideals and invariant additive subgroups in rings

Abstract We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic where every additive commutator is a sum of square‐zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero‐product balanced algebras.