Semiderivations Satisfying Certain Algebraic Identities on Jordan Ideals

Vincenzo De Filippis,Abdellah Mamouni,Lahcen Oukhtite

doi:10.1155/2013/738368

Abstract

Let be a ring. An additive mapping is called semiderivation of if there exists an endomorphism of such that and , for all in . Here we prove that if is a 2-torsion free -prime ring and a nonzero -Jordan ideal of such that for all , then either is commutative or for all . Moreover, we initiate the study of generalized semiderivations in prime rings.

Highlights

Introduction and PreliminariesThroughout this paper, R will denote an associative ring with center Z(R)
If Ro denotes the opposite ring of a prime ring R, R × Ro equipped with the exchange involution ∗ex, defined by ∗ex(x, y) = (y, x), is ∗ex-prime but not prime
In case R is a prime ring and d ≠ 0 is a semiderivation with associated function g, Chang ([12], Theorem 1) has shown that g must necessarily be a ring endomorphism

Summary

Introduction

Introduction and PreliminariesThroughout this paper, R will denote an associative ring with center Z(R). We prove that if R is a 2-torsion free ∗-prime ring and J a nonzero ∗-Jordan ideal of R such that d([x, y]) − [x, y] = 0 for all x, y ∈ J, either R is commutative or d(x) = x − g(x) for all x ∈ R.

Results

Conclusion