All-derivable points in the algebra of all upper triangular matrices (Q932149)

Let \(\mathcal{TM}_n\) be the algebra of all \(n\times n\) upper triangular matrices, and let \(L(\mathcal{TM}_n)\) denote the set of all linear mappings on \(\mathcal{TM}_n\). For \(\varphi\in L(\mathcal{TM}_n)\), \(\varphi\) is said to be a derivation if \(\varphi(ST)=\varphi(S)T+S\varphi(T)\) for any \(S,T\in\mathcal{TM}_n\). Fix a matrix \(G\in\mathcal{TM}_n\), then \(\varphi\in L(\mathcal{TM}_n)\) is said to be a derivable mapping at \(G\) if \(\varphi(ST)=\varphi(S)T+S\varphi(T)\) for any \(S,T\in\mathcal{TM}_n\) with \(ST=G\). In this paper, the authors prove that \(G\in\mathcal{TM}_n\) is an all derivable point of \(\mathcal{TM}_n\) if and only if \(G\neq0\).