Fully noncentral Lie ideals and invariant additive subgroups in rings

TL;DR

This paper establishes conditions under which fully noncentral Lie ideals and invariant additive subgroups in rings contain all additive commutators, with applications to certain algebraic structures like zero-product balanced algebras.

Contribution

It characterizes when fully noncentral subgroups are Lie ideals or invariant, extending understanding of their structure in rings and algebras.

Findings

Fully noncentral subgroups contain all additive commutators.

In unital algebras over fields with characteristic not 2, such subspaces are Lie ideals iff invariant under inner automorphisms.

Application to zero-product balanced algebras.

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 $\neq 2$ 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.