Generalized Jordan derivations of Incidence Algebras

TL;DR

This paper extends the understanding of derivations in incidence algebras by showing that generalized Jordan derivations are actually generalized derivations over 2-torsion free rings, generalizing previous results.

Contribution

It proves that generalized Jordan derivations of incidence algebras are generalized derivations when the ring is 2-torsion free, broadening the class of derivations characterized.

Findings

Generalized Jordan derivations are generalized derivations over 2-torsion free rings.

The result generalizes Xiao's theorem on Jordan derivations.

Provides a broader understanding of derivation structures in incidence algebras.

Abstract

For a given ring $\mathfrak{R}$ and a locally finite pre-ordered set $(X, \leq)$, consider $I(X, \mathfrak{R})$ to be the incidence algebra of $X$ over $\mathfrak{R}$. Motivated by a Xiao's result which states that every Jordan derivation of $I(X,\mathfrak{R})$ is a derivation in the case $\mathfrak{R}$ is $2$-torsion free, one proves that each generalized Jordan derivation of $I(X,\mathfrak{R})$ is a generalized derivation provided $\mathfrak{R}$ is $2$-torsion free, getting as a consequence the above mentioned result.