Jordan Derivations of Incidence Algebras

TL;DR

This paper characterizes derivations of incidence algebras over commutative rings and proves that Jordan derivations are derivations when the ring is 2-torsion free, advancing understanding of algebraic structures.

Contribution

It provides a complete characterization of derivations and shows Jordan derivations coincide with derivations under specific conditions in incidence algebras.

Findings

All derivations of incidence algebras are characterized.

Every Jordan derivation is a derivation if the ring is 2-torsion free.

The results extend the understanding of algebraic derivations in incidence algebras.

Abstract

Let $\mathcal{R}$ be a commutative ring with identity, $I(X,\mathcal{R})$ be the incidence algebra of a locally finite pre-ordered set $X$. In this note, we characterise the derivations of $I(X,\mathcal{R})$ and prove that every Jordan derivation of $I(X,\mathcal{R})$ is a derivation provided that $\mathcal{R}$ is $2$-torsion free.