Automorphisms and derivations of finite-dimensional algebras (Q2118932)

In this article, the author considers a finite-dimensional algebra over the field of characteristic not equal to \(2\). An attempt to demonstrate that an inner derivation and a linear map whose image resides in the radical of \(A\) combine to form a linear map \(D: A\to A\) fulfilling \(xD(x)x\in [A, A]\) for every \(x\in A\). A linear map \(T: A\to A\) satisfies \(T(x)^3-x^3\in[A, A]\) for all \(x\in A\) if and only if there exists a Jordan automorphism \(J\) of \(A\) residing in the multiplication algebra of \(A\) and a central element \(\alpha\) fulfilling \(\alpha^3 = 1\) such that \(T(x) = \alpha J(x)\) for all \(x\in A\). This study is further applied to the local derivations and local (Jordan) automorphisms. All the content is correct according to my knowledge, well written and thoroughly explained making it easy to understand.