Jordan derivations of finitary incidence rings

TL;DR

This paper establishes conditions under which all Jordan derivations of finitary incidence rings are actual derivations, extending previous results and including matrix rings over rings with more than one element.

Contribution

It provides a new criterion for Jordan derivations to be derivations in finitary incidence rings and generalizes existing theorems to matrix rings over rings.

Findings

Jordan derivations of finitary incidence rings are derivations under certain conditions

Each Jordan derivation of matrix rings over rings with more than one element is a derivation

Generalization of previous theorems on derivations in algebraic structures

Abstract

Let $P$ be a preordered set, $R$ a ring and $FI(P,R)$ the finitary incidence ring of $P$ over $R$. We find a criterion for all Jordan derivations of $FI(P,R)$ to be derivations and generalize Theorem 3.3 from arXiv:1411.6123. In particular, we prove that each Jordan derivation of the ring $RFM_I(R)$ of row-finite $I\times I$-matrices over $R$ is a derivation, if $|I|>1$.