Differential identities with automorphisms and antiautomorphisms. I
This paper generalizes a result of \textit{V. K. Kharchenko} [Algebra Logika 17, 220-238 (1978; Zbl 0423.16011)] on differential identities of prime rings, by allowing anti-automorphisms, as well as derivations and automorphisms, to act on the variables. The author defines the two-sided Utumi quotient ring \(U\) of a prime ring \(R\) with extended centroid \(C\), and defines generalized polynomials with coefficients in \(U\) to be the elements of \(U*_ C C\{X\}\), the free product over \(C\) of \(U\) and the free algebra \(C\{X\}\). Derivations, automorphisms, and anti-automorphisms of \(R\) extend to \(U\). If \(w(j)\) is an endomorphism of \(U\) given as a word in derivations, automorphisms, and anti-automorphisms of \(R\), then an expression \(f(x^{w(j)}_ i)\) is an identity for \(R\) if \(f(x_{ij})\) is a generalized polynomial with coefficients in \(U\) and \(f(r^{w(j)}_ i)=0\) for all substitutions, where \(r^{w(j)}_ i\) is the image of \(r_ i\in R\) under \(w(j)\). The main result is that if \(R\) satisfies a nontrivial such identity, then \(R\) must satisfy a nontrivial generalized polynomial.
scientific article; zbMATH DE number 63618
scientific article