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Fully noncentral Lie ideals and invariant additive subgroups in rings

Eusebio Gardella, Tsiu-Kwen Lee, Hannes Thiel

TL;DR

The paper investigates when a fully noncentral additive subgroup of a ring must be a Lie ideal, linking this property to invariance under inner automorphisms. It develops a general framework using the normalizer-type operator $T(V)$ and its iterates to derive criteria ensuring $[R,R]\subseteq V$ and the Lie-ideal structure, under hypotheses such as $ ext{char}$ not equal to $2$ and commutators decomposable into square-zero elements or nilpotent elements, plus non-exceptional prime-ideal conditions. It then specializes to rings where every proper ideal lies in a prime (or non-exceptional prime) ideal, establishing equivalences that tie fullness of commutator-derived subgroups to the inclusion $[R,R]\subseteq L$ and to $R=L^2$, thereby clarifying when fully noncentral subspaces are forced to be Lie ideals. Further, the paper connects these algebraic insights to zero-product balanced algebras, showing that in many important classes, fully noncentral subspaces invariant under inner automorphisms are precisely Lie ideals with $[A,A]\subseteq V$, and explores the description of $[R,R]$ as sums of square-zero elements, including explicit structure results and counterexamples. The work advances understanding of how inner automorphism symmetries interact with the Lie structure in broad ring-theoretic contexts, with potential implications for operator algebras and PI/ring theory.

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.

Fully noncentral Lie ideals and invariant additive subgroups in rings

TL;DR

The paper investigates when a fully noncentral additive subgroup of a ring must be a Lie ideal, linking this property to invariance under inner automorphisms. It develops a general framework using the normalizer-type operator and its iterates to derive criteria ensuring and the Lie-ideal structure, under hypotheses such as not equal to and commutators decomposable into square-zero elements or nilpotent elements, plus non-exceptional prime-ideal conditions. It then specializes to rings where every proper ideal lies in a prime (or non-exceptional prime) ideal, establishing equivalences that tie fullness of commutator-derived subgroups to the inclusion and to , thereby clarifying when fully noncentral subspaces are forced to be Lie ideals. Further, the paper connects these algebraic insights to zero-product balanced algebras, showing that in many important classes, fully noncentral subspaces invariant under inner automorphisms are precisely Lie ideals with , and explores the description of as sums of square-zero elements, including explicit structure results and counterexamples. The work advances understanding of how inner automorphism symmetries interact with the Lie structure in broad ring-theoretic contexts, with potential implications for operator algebras and PI/ring theory.

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.
Paper Structure (6 sections, 27 theorems, 58 equations)

This paper contains 6 sections, 27 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Results about general rings
  3. Rings where proper ideals are contained in prime ideals
  4. Rings where proper ideals are contained in non-exceptional prime ideals
  5. Fully noncentral, invariant subspaces
  6. Commutators as sums of square-zero elements

Key Result

Theorem 1

Let $A$ be an algebra over a field $\neq \{0,1\}$ such that every commutator in $A$ is a sum of square-zero elements, and every proper ideal is contained in a non-exceptional prime ideal. Let $V \subseteq A$ be a fully noncentral subspace. Then the following are equivalent:

Theorems & Definitions (61)

  • Definition 1
  • Theorem 1
  • Lemma 2.1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2.2
  • proof
  • ...and 51 more