Jordan Higher Derivable Mappings on Rings

Mohammad Ashraf,Nazia Parveen

doi:10.1155/2014/672387

Abstract

Let R be a ring. We say that a family of maps D={dn}n∈N is a Jordan higher derivable map (without assumption of additivity) on R if d0=IR (the identity map on R) and dn(ab+ba)=∑p+q=n‍dp(a)dq(b)+∑p+q=n‍dp(b)dq(a) hold for all a,b∈R and for each n∈N. In this paper, we show that every Jordan higher derivable map on a ring under certain assumptions becomes a higher derivation. As its application, we get that every Jordan higher derivable map on Banach algebra is an additive higher derivation.

Highlights

Characterizing the interrelation between the multiplicative and the additive structures of a ring is an interesting topic and has received attention of many mathematicians. It is a well-known result due to Martindale III [1] that every multiplicative bijective map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is necessarily additive
In the present paper we introduce the notion of higher derivable map on R and provide a sufficient condition on a ring R under which a Jordan higher derivable map becomes a higher derivation
In order to illustrate the results in the literature focused on derivable maps and higher derivation we will introduce the concept of higher derivable maps on certain classes of rings

Summary

Introduction

An additive mapping d : R → R is said to be a derivation (resp., Jordan derivation) if d(ab) = d(a)b + ad(b) (resp., d(a2) = d(a)a + ad(a)) holds for any a, b ∈ R. Characterizing the interrelation between the multiplicative and the additive structures of a ring is an interesting topic and has received attention of many mathematicians It is a well-known result due to Martindale III [1] that every multiplicative bijective map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is necessarily additive. Lu [2] proved that “each derivable map on a 2-torsion free unital prime ring containing a nontrivial idempotent is a derivation.”. In the present paper we introduce the notion of higher derivable (resp., Jordan higher derivable) map on R and provide a sufficient condition on a ring R under which a Jordan higher derivable map becomes a higher derivation

Additivity of Jordan Higher Derivable Mappings

Conflict of Interests