Higher derivations and jordan higher derivations of Г-rings

Abstract

Let M and Г be additive abelian groups. If there exists a map ⋅ sending M x Г x M into M, denote the image of a ⋅ γ ⋅ b by, simply, aγb for all a, b ϵ M and γ ϵ Г, satisfying the following properties: for each a, b, c ϵ M and γ, β ϵ Г, (i) (aγb)βc = aγ(bβc); (ii) (a + b)γc = aγc + bγc, a(γ + β)c = aγc + aβc and aγ(b + c) = aγb + aγc, then M is called a Г-ring. Let M be a Г-ring and U be an ideal of M. We introduce the concept of higher derivations, Jordan higher derivations of M, higher derivations, Jordan higher derivations of U into M and generalized higher derivations, Jordan generalized higher derivations of M. Our main interests are finding appropriate conditions for a Г-ring in order to obtain that 1. Jordan higher derivations and higher derivations of a Г-ring are the same, 2. Jordan higher derivations and higher derivations of an ideal of a Г-ring, are identical, and 3. Jordan generalized higher derivations and generalized higher derivations of a Г-ring are coincide.