Commuting Jordan Derivations on Triangular Rings Are Zero

The main purpose of this article is to show that every commuting Jordan derivation on triangular rings (unital or not) is identically zero. Using this result, we prove that if $\mathcal{A}$ is a $2$-torsion free ring that is either semiprime or satisfies Condition (P), then, under certain conditions, every commuting Jordan derivation of $\mathcal{A}$ into itself is identically zero.