[Paper Review] Commutativity conditions on derivations and Lie ideals ins-prime rings

Lahcen Oukhtite, S. Salhi+1 more|arXiv (Cornell University)|Jan 1, 2010

Advanced Topics in Algebra13 references3 citations

TL;DR

This paper investigates commutativity conditions for derivations on $σ$-prime rings, focusing on square-closed $σ$-Lie ideals. It proves that if a derivation $d$ centralizes such a Lie ideal $U$, or satisfies certain commutativity conditions on $U$, then either $d = 0$ or $U$ is contained in the center of $R$, establishing strong structural constraints on derivations in this setting.

ABSTRACT

Let $R$ be a 2-torsion free $\sigma$-prime ring, $U$ a nonzero square closed $\sigma$-Lie ideal of $R$ and let $d$ be a derivation of $R$. In this paper it is shown that: 1) If $d$ is centralizing on $U$, then $d = 0$ or $U \subseteq Z(R)$. 2) If either $d([x, y]) = 0$ for all $x, y \in U$, or $[d(x), d(y)] = 0$ for all $x, y \in U$ and $d$ commutes with $\sigma$ on $U$, then $d = 0$ or $U \subseteq Z(R)$.

Motivation & Objective

• To explore the behavior of derivations on $σ$-prime rings under specific commutativity constraints.
• To determine when a derivation that centralizes or commutes with elements of a $σ$-Lie ideal must vanish or force the ideal into the center of the ring.
• To extend known commutativity results from prime rings to the more general setting of $σ$-prime rings with $σ$-Lie ideals.
• To analyze the interplay between derivations, automorphisms ($σ$), and Lie ideal structure in noncommutative rings.
• To establish sufficient conditions under which derivations are trivial or ideals are central in $σ$-prime rings.

Proposed method

• Use of the 2-torsion free assumption to simplify algebraic identities involving derivations and commutators.
• Application of the $σ$-prime ring property to restrict the existence of nontrivial ideals and derivations.
• Analysis of derivations acting on square-closed $σ$-Lie ideals $U$ via commutator identities and centralizing conditions.
• Employment of the condition that $d$ commutes with $σ$ on $U$ to derive structural constraints on $d$ and $U$.
• Use of the identity $d([x,y]) = 0$ or $[d(x), d(y)] = 0$ for all $x,y \in U$ to deduce that $d$ must be trivial or $U \subseteq Z(R)$.
• Leveraging Lie ideal properties and the primeness of $R$ under $σ$ to eliminate non-central solutions.

Experimental results

Research questions

• RQ1 Under what conditions does a derivation $d$ that centralizes a square-closed $σ$-Lie ideal $U$ in a 2-torsion free $σ$-prime ring $R$ imply $d = 0$ or $U \subseteq Z(R)$?
• RQ2 What happens when $d([x,y]) = 0$ for all $x,y \in U$ in such a ring? Does this force $d = 0$ or $U \subseteq Z(R)$?
• RQ3 If $[d(x), d(y)] = 0$ for all $x,y \in U$ and $d$ commutes with $\sigma$ on $U$, does this imply $d = 0$ or $U \subseteq Z(R)$?
• RQ4 How does the $σ$-prime condition interact with derivations and Lie ideals to constrain the structure of $R$?
• RQ5 Can nontrivial derivations exist on non-central $σ$-Lie ideals in 2-torsion free $σ$-prime rings under these commutativity conditions?

Key findings

• If $d$ is centralizing on $U$, then either $d = 0$ or $U$ is contained in the center $Z(R)$ of the ring $R$.
• If $d([x,y]) = 0$ for all $x,y \in U$, then $d = 0$ or $U \subseteq Z(R)$, under the 2-torsion free and $σ$-prime assumptions.
• If $[d(x), d(y)] = 0$ for all $x,y \in U$ and $d$ commutes with $\sigma$ on $U$, then $d = 0$ or $U \subseteq Z(R)$.
• The results hold specifically in 2-torsion free $σ$-prime rings, which ensures the validity of key algebraic manipulations.
• The structure of $U$ as a square-closed $σ$-Lie ideal is essential in deriving the conclusions, as it enables the use of Lie ideal identities.
• The central conclusion is that under the given conditions, derivations are either trivial or force the Lie ideal to be central, indicating strong rigidity in the ring's structure.