Generalized Jordan derivations on semiprime rings

TL;DR

This paper proves that under mild conditions, every additive generalized Jordan derivation on a semiprime ring with a specific idempotent element is actually a generalized derivation, clarifying the structure of such derivations.

Contribution

It establishes that generalized Jordan derivations on certain semiprime rings are equivalent to generalized derivations, extending understanding of derivation structures.

Findings

Every additive generalized Jordan derivation is a generalized derivation.

The result applies to semiprime rings with a nontrivial idempotent element.

Conditions are mild and do not require strong restrictions on the ring.

Abstract

The purpose of this note is to prove the following. Suppose $\R$ is a semiprime unity ring having an idempotent element e $\left(e \neq 0, e \neq 1\right)$ which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on $\R$ is a generalized derivation.