Generalized Jordan triple derivations associated with Hochschild 2\u2013cocycles of rings

O H Ezzat,H Nabiel

doi:10.1186/s42787-019-0003-3

Abstract

In the present work, we introduce the notion of a generalized Jordan triple derivation associated with a Hochschild 2–cocycle, and we prove results which imply under some conditions that every generalized Jordan triple derivation associated with a Hochschild 2–cocycle of a prime ring with characteristic different from 2 is a generalized derivation associated with a Hochschild 2–cocycle.

Highlights

Let R denote an associative ring with center Z(R)
An additive map f : R → M is called a generalized derivation associated with a Hochschild 2–cocycle α if f = f (x)y + xf (y) + α(x, y) for all x, y ∈ R, and f is called a generalized Jordan derivation associated with α if f (x2) = f (x)x + xf (x) + α(x, x) for all x ∈ R
We introduce the notion of generalized Jordan triple derivations associated with Hochschild 2–cocycles in the following way

Summary

Introduction

Let R denote an associative ring with center Z(R). A ring R is said to have characteristic n if n is the least positive integer such that nx = 0 for all x ∈ R, and of characteristic not n if nx = 0, x ∈ R, x = 0. An additive map f : R → M is called a generalized derivation associated with a Hochschild 2–cocycle α if f (xy) = f (x)y + xf (y) + α(x, y) for all x, y ∈ R, and f is called a generalized Jordan derivation associated with α if f (x2) = f (x)x + xf (x) + α(x, x) for all x ∈ R.

Results

Conclusion