ON 4-PERMUTING 4-DERIVATIONS IN PRIME AND SEMIPRIME RINGS

Abstract

Let R be a 2-torsion free semiprime ring. Suppose that there exists a 4-permuting 4-derivation ${\Delta}:R{\times}R{\times}R{\times}R{\rightarrow}R$ such that the trace is centralizing on R. Then the trace of ${\Delta}$ is commuting on R. In particular, if R is a 3!-torsion free prime ring and ${\Delta}$ is nonzero under the same condition, then R is commutative.

Keywords

• prime ring;
• semiprime ring;
• commuting map;
• centralizing map;
• n-permuting map;
• derivation;
• 4-derivation