# Permuting tri-derivations in prime and semi-prime rings

• H. Durna Cumhuriyet University
• S. OĞUZ Cumhuriyet University

### Abstract

Let $$R$$ be a ring and $$U\neq0$$ be a square closed Lie ideal of $$R$$. A tri-additive permuting map $$D:R\times R\times R\rightarrow R$$ is called permuting tri-derivation if, for any $$y,z\in R$$, the map $$x\mapsto D(x,y,z)$$ is a derivation. A mapping $$d:R\rightarrow R$$ defined by $$d(x)=D(x,x,x)$$ is called the trace of $$D$$. In the present paper, we show that $$U\subseteq Z$$ such that $$R$$ is a prime and semi-prime ring admitting the trace $d$ satisfying the several conditions of permuting tri-derivation.