Characterizations of derivations on incidence algebras by local actions

Abstract

Let $(X,\le )$ be a locally finite pre-ordered set and $\mathcal{R}$ be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra $I(X,\mathcal{R})$ which is derivable at zero is a generalized derivation and every local derivation on $I(X,\mathcal{R})$ is a derivation.

Keywords:

• Incidence algebra
• local derivation
• map derivable at zero
• zero product determined algebra

2020 Mathematics Subject Classification:

• Primary: 16W25
• Secondary: 16R60
• 47L35

Acknowledgments

We are sincerely thankful to the referees for their careful reading of the manuscript and their valuable comments and suggestions.

Additional information

Funding

The first author is supported by the NSFC (No. 12201224). The second author is supported by the NSF of Fujian Province (No. 2023J01126).