On (m, n)-derivations of Some Algebras

TL;DR

This paper characterizes ( extit{m, n})-derivable mappings at specific points on generalized matrix and CSL algebras, and explores their connections with Lie, Jordan, and standard derivations.

Contribution

It provides a comprehensive characterization of ( extit{m, n})-derivable mappings at key points on certain algebras, linking them to known derivation types.

Findings

Characterization of ( extit{m, n})-derivable mappings at 0, $I_ ext{A}igoplus0$, and $I$ on generalized matrix algebras.

Analysis of ( extit{m, n})-derivable mappings at 0 on CSL algebras.

Established relationships between ( extit{m, n})-derivable mappings and Lie, Jordan, and standard derivations.

Abstract

Let $\mathcal{A}$ be a unital algebra, $\delta$ be a linear mapping from $\mathcal{A}$ into itself and $m$, $n$ be fixed integers. We call $\delta$ an (\textit{m, n})-derivable mapping at $Z$, if $m\delta(AB)+n\delta(BA)=m\delta(A)B+mA\delta(B)+n\delta(B)A+nB\delta(A)$ for all $A, B\in \mathcal{A}$ with $AB=Z$. In this paper, (\textit{m, n})-derivable mappings at 0 (resp. $I_\mathcal{A}\oplus0$, $I$) on generalized matrix algebras are characterized. We also study (\textit{m, n})-derivable mappings at 0 on CSL algebras. We reveal the relationship between this kind of mappings with Lie derivations, Jordan derivations and derivations.