On the trace of permuting tri-derivations on rings

Abstract

In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring.
Let $R$ be a $2,3$-torsion free prime ring and $F:R\times R\times R\rightarrow R$ be a permuting tri-derivation with trace $f$, $d:R\rightarrow R$ be a derivation. In particular, the following assertions have been proved:
1) if $[d(r),r]=f(r)$ for all $r\in R$, then $R$ is commutative or $d=0$ (Theorem 1);\

2) if $g:R\rightarrow R$ is an endomorphism such that $F(d(r),r,r)=g(r)$ for all $r\in R$, then $F=0$ or $d=0$ (Theorem 2);

3) if $F(d(r),r,r)=f(r)$ for all $r\in R$, then $(i)$ $F=0$ or $d=0$, $(ii)$ $d(r)\circ f(r)=0$ for all $r\in R$ (Theorem 3).

In the other hand, if there exist permuting tri-derivations $F_{1},F_{2}:R\times R\times R\rightarrow R$ such that $F_{1}(f_{2}(r),r,r)=f_{1}(r)$ for all $r\in R$, where $f_{1}$ and $%f_{2}$ are traces of $F_{1}$ and $F_{2}$, respectively, then $(i)$ $F_{1}=0$ or $F_{2}=0$, $(ii)$ $f_{1}(r)\circ f_{2}(r)=0$ for all $r\in R$ (Theorem 4).