# On Generalized $(m, n)$-Jordan Derivations and Centralizers of Semiprime Rings

## TL;DR

This paper confirms two conjectures by proving that in semiprime rings, generalized $(m,n)$-Jordan derivations and centralizers are actually derivations and centralizers, under certain conditions, clarifying their structure.

## Contribution

It provides an affirmative proof that generalized $(m,n)$-Jordan derivations and centralizers are standard derivations and centralizers in semiprime rings, resolving previous conjectures.

## Key findings

- Generalized $(m,n)$-Jordan derivations are derivations under certain conditions.
- Generalized $(m,n)$-Jordan centralizers are two-sided centralizers under certain conditions.
- Results apply to semiprime rings with specific torsion restrictions.

## Abstract

In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fo\v{s}ner, \textit{A note on generalized $(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivation (resp., a generalized $(m,n)$-Jordan centralizer) is a derivation (resp., a two-sided centralizer).