Jordan derivations and antiderivations on triangular matrices (Q1774972)

Let \(\mathcal{C}\) be a commutative ring with 1, \(\mathcal{T}_{n}\) be the ring of all upper triangular matrices over \(\mathcal{C}\) and \(\mathcal{M}\) be a \(\mathcal{T}_{n}\)-bimodule. Then a \(\mathcal{C}\)-linear map \(\Delta :\mathcal{T}_{n}\rightarrow\mathcal{M}\) is a Jordan derivation if \(\Delta(ab+ba)=\Delta(a)b+a\Delta(b)+\Delta(b)a+b\Delta(a)\) for all \(a,b\). The author proves that every Jordan deriviation \(\Delta\) is uniquely expressible as the sum of a derivation and an antiderivation (that is, a \(\mathcal{C} \)-linear map \(\delta\) such that \(\delta(ab)=\delta(b)a+b\delta(a)\)) which vanishes on the diagonal matrices. No such decompositon exists for the free algebra \(\mathcal{C}\left\langle x,y\right\rangle \) in noncommuting variables \(x,y\). As a corollary, the author deduces that, for \(n\geq2\), every Jordan derivation of \(\mathcal{T}_{n}\) into itself is a derivation.