Prime rings having nontrivial centralizers of (skew) traces of Lie ideals

TL;DR

This paper investigates the structure of prime rings with involution, showing that certain derivations vanish on specific trace subgroups, leading to the centrality of their squares, and generalizes recent results on simple artinian rings.

Contribution

It establishes new conditions under which trace-related subgroups in prime rings are central, extending previous results to broader classes of rings with involution.

Findings

If a derivation annihilates the trace of a Lie ideal, then the square of the trace subgroup is central.

The results apply to subrings of division rings invariant under automorphisms.

Generalizes recent theorems on simple artinian rings with involution.

Abstract

Let $R$ be a prime ring with center $Z(R)$ and with involution $*$. Given an additive subgroup $A$ of $R$, let $T(A):=\{x+x^*\mid x\in A\}$ and $K_0(A):=\{x-x^*\mid x\in A\}$. Let $L$ be a non-abelian Lie ideal of $R$. It is proved that if $d$ is a nonzero derivation of $R$ satisfying $d(T(L))=0$ (resp. $d(K_0(L))=0$), then $T(R)^2\subseteq Z(R)$ (resp. $K_0(R)^2\subseteq Z(R)$). These results are applied to the study of $d(T(M))=0$ and $d(K_0(M))=0$ for noncentral $*$-subrings $M$ of a division ring $R$ such that $M$ is invariant under all inner automorphisms of $R$, and for noncentral additive subgroups $M$ of a prime ring $R$ containing a nontrivial idempotent such that $M$ is invariant under all special inner automorphisms of $R$. The obtained theorems also generalize some recent results on simple artinian rings with involution due to M. Chacron.