Lie (Jordan) derivations of arbitrary triangular algebras
The author constructs a triangular algebra from a given triangular algebra, using the notion of maximal left (right) ring of quotients (Theorem 2.1: Let $U=\text{Tri}(A, M,B)$ be a triangular algebra. Set $U^0$ as $2\times 2$ a matrix such that $a_{11}= A((\tau_r^{-1}) Z(B))$, $a_{12}=M$, $a_{21}=0$, $a_{22}=B((\tau_l )Z(A))$. Then $U^0$ is a triangular algebra such that $U$ is a subalgebra of $U^0$ having the same unity, where $\tau$ is a unique algebra isomorphism). Also, the author has a description of Lie derivations on arbitrary triangular algebras through the constructed triangular algebra (Theorem 3.1: Let $U=\text{Tri}(A, M, B)$ be a triangular algebra. Let $L$ be a Lie derivation on $U$. Then there exists a triangular algebra $U^0$ such that $U$ is a subalgebra of $U^0$ having the same unity and $L$ can be written as $L=\delta +\tau$, where $\delta: U\to U^0$ is a derivation and $\tau: U\to Z( U^0)$ is a linear map such that $\tau ([x, y]) = 0$ for all $x, y\in U$). Consequently, the author obtains a description of Jordan derivations on arbitrary triangular algebras (Corollary 3.1. Let $U =\text{Tri} (A, M, B)$ be a triangular algebra. Let $D$ be a Jordan derivation on $U$. Suppose that $U$ is 2-torsion free. Then $D$ is a derivation. Otherwise, there exists a triangular algebra $U^0$ such that $U$ is a subalgebra of $U^0$ having the same unity and $D$ can be written as $D=\delta+\tau$, where $\delta:U\to U^0$ is a derivation and $\tau: U\to Z (U^0)$ is a linear map such that $\tau(x\circ y) =0$ for all $x, y\in U$). Finally, the author presents other results like propositions, lemmas and corollaries which help him to reach his aim.
scientific article; zbMATH DE number 7135054
scientific article