Notes on generalized \((\sigma,\tau)\)-derivation.
Let \(R\) be a prime ring with automorphisms \(\sigma\) and \(\tau\), \((\sigma,\tau)\)-derivation \(d\), and generalized \((\sigma,\tau)\)-derivation \((f,d)\). The main results in the paper claim that when \(f\) acts as a homomorphism, or as an antihomomorphism, then \(d=0\), and that for \(f\) acting as a homomorphism on a Lie ideal \(U\) of \(R\), then either \(U\) is central or \(d=0\). Theorem 2, when \(f\) is a homomorphism, is incorrect as stated, so leads to an incorrect statement for Theorem 3 about antihomomorphisms, and the same problem holds for Theorem 5, which is Theorem 2 for Lie ideals. The error in the proof of these results is the authors' assumption that \(\tau-\sigma\) is a homomorphism, which cannot be true unless \(\tau=\sigma\). The authors fail to realize that for any \(\sigma\) and \(\tau\), \(d=\tau-\sigma\) is a \((\sigma,\tau)\)-derivation and \((\tau,\tau-\sigma)\) is a generalized \((\sigma,\tau)\)-derivation with \(\tau\) a homomorphism. Finally, the proof of Theorem 6 requires the assumption that \(U\) is closed under squares, so contains \(2uv\) for \(u,v\in U\).
scientific article; zbMATH DE number 5780247