Characterization of Jordan centralizers and Jordan two-sided centralizers on triangular rings without assuming unity

TL;DR

This paper proves that Jordan centralizers and Jordan two-sided centralizers are actually centralizers on triangular rings without the need for a unity assumption, and explores related derivation properties.

Contribution

It establishes that Jordan centralizers and Jordan two-sided centralizers coincide with centralizers on triangular rings without assuming the existence of a unity element.

Findings

Jordan centralizers are centralizers on triangular rings

Jordan two-sided centralizers are centralizers on triangular rings

Jordan generalized derivations are two-sided generalized derivations

Abstract

The main purpose of this paper is to show that every Jordan centralizer and every Jordan two-sided centralizer is a centralizer on triangular rings without assuming unity. As an application, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. Some other related results are also discussed.