On some equations related to derivations in rings

Joso Vukman,Irena Kosi-Ulbl

doi:10.1155/ijmms.2005.2703

Abstract

Letmandnbe positive integers withm+n≠0, and letRbe an(m+n+2)!-torsion free semiprime ring with identity element. Suppose there exists an additive mappingD:R→R, such thatD(xm+n+1)=(m+n+1)xmD(x)xnis fulfilled for allx∈R, thenDis a derivation which mapsRinto its center.

Highlights

2704 On some equations related to derivations in rings
Let us recall that any Jordan derivation on a 2-torsion free semiprime ring is a derivation
It is well known and easy to prove that any commuting derivation on a semiprime ring R maps R into Z(R)

Summary

Introduction

Let us point out that any commuting derivation on an arbitrary ring satisfies the relation D(x3) = 3xD(x)x. 2704 On some equations related to derivations in rings N with m ≥ 0, n ≥ 0, and m + n = 0, let R be an (m + n + 2)!torsion free semiprime ring with identity element. In case m = 1, n = 0 (we adopt the convention x0 = e, for all x ∈ R, where e denotes the identity element), we have an additive mapping satisfying the relation D(x2) = 2xD(x), x ∈ R.

Results

Conclusion