Generalized (m,n)-Jordan centralizers and derivations on semiprime rings

TL;DR

This paper proves two longstanding conjectures from 2013 and 2014 concerning generalized (m,n)-Jordan centralizers and derivations on semiprime rings, advancing the understanding of ring structure and operator behavior.

Contribution

It confirms two conjectures related to generalized (m,n)-Jordan centralizers and derivations on semiprime rings, providing new insights into their properties.

Findings

Conjecture 1 by Fosner proved in 2013

Conjecture 1 by Ali and Fosner proved in 2014

Enhanced understanding of generalized (m,n)-Jordan structures

Abstract

In this article, we prove Conjecture $1$ posed in 2013 by Fo$\check{s}$ner \cite{fos} and Conjecture $1$ posed in 2014 by Ali and Fo$\check{s}$ner \cite{ali} related to generalized $(m,n)$-Jordan centralizer and derivation respectively.