Jordan Left Derivations of Two Torsion Free ГM – Modules

Abstract

Let M be a Г-ring and X be a 2-torsionfree left ГM-module. The purpose of this paper is to investigate Jordan left derivations on M considering aαbβc=aβbαc, for every a,b,c∈M and α,β∈Г . We show that the existence of a nonzero Jordan left derivation of M into X implies M is commutative. We also show that if X = M is a semiprime Г-ring, then the derivation is a mapping from M into its centre. Finally we show that if M is a prime Г- ring, then every Jordan left derivation d: M→ M is a left derivation.