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