Identities with skew derivations
In this long and complex but well written paper the author extends the seminal work of \textit{V. K. Kharchenko} [Algebra Logika 17, 220-238 (1978; Zbl 0423.16011), and ibid. 18, 86-119 (1979; Zbl 0464.16027)] to skew derivations of prime rings \(R\). Roughly speaking a differential polynomial is a generalized polynomial with variables \(x^H_j\) for \(H\) a composition of derivations (and (anti) automorphisms) of \(R\). Substitution of \(r_j\in R\) for \(x^H_j\) yields \(H(r_j)\). A differential identity for \(R\) is a differential polynomial so that all evaluations result in zero. The approach of the author extends the work of Kharchenko by allowing exponents of the variables to be compositions including skew derivations and more generally \((g,h)\) derivations or higher derivations. When \(g\) and \(h\) are automorphisms of \(R\) a \((g,h)\) derivation \(D\) is an additive endomorphism of \(R\) so that \(D(ab)=g(a)D(b)+ D(a)h(b)\). The aim of the paper is to show that when \(R\) satisfies an identity with these general exponents then \(R\) satisfies a generalized polynomial identity. Much of the paper concerns the definition and identification of the trivial identities, essentially those which hold by the definitions of the maps under consideration. A very simple example of a trivial identity would be \(x^E-x^{DF}+x^{FD}\) where \(D\) and \(F\) are derivations of \(R\) and \(E\) is the derivation \(DF-FD\).
scientific article; zbMATH DE number 1428244
scientific article