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Differential identities with automorphisms and antiautomorphisms. II. (English) Zbl 0793.16014
[Part I, cf. ibid. 149, No. 2, 371-404 (1991; Zbl 0773.16007).]
The author considers identities of prime rings which have derivations, automorphisms, and anti-automorphisms acting. For a prime ring \(R\), let \(C\) be its extended centroid and \(U\) its two-sided Utumi quotient ring. Let \(D(R)\) be the set of derivations of \(R\), and \(G\) the group of automorphisms and anti-automorphisms of \(R\). An element \(g\in G\) is Frobenius if for all \(c\in C\), \(c^ g=c\) when \(\text{char}(R)=0\), and \(c^ g=c^ n\) for some fixed \(n\in Z\) when \(\text{char}(R)=p > 0\). The normal subgroup of Frobenius elements of \(G\) is \(G_ F\). Any identity of \(R\) having elements of \(D(R)\) and \(G\) acting is equivalent modulo “universal identities” to an identity \(\Phi(x_ i^{\Delta_ j g_ k f_ t})= \Psi(x_{ijkt})\in U *_ C C\{X\}\), the free product over \(C\) of \(U\) and the free algebra \(C\{X\}\). Here \(\Delta_ j\) is a “regular” word of outer derivations, \(g_ k\in G - G_ F\), and \(f_ k\in G_ F\), where \(\{g_ k\}\) represent distinct cosets of \(G_ F\) in \(G\), and \(\{f_ t\}\) represent distinct cosets of the subgroup of inner automorphisms in \(G_ F\). The main result shows that if \(\Phi(x_ i^{\Delta_ j g_ k f_ t})\) is an identity for a nonzero ideal of \(R\), then \(\Psi(x_{ijk}^{f_ t})\) is an identity for \(U\). If no anti-automorphisms act, then \(\Psi(x_{ijk})\) is an identity for the left Utumi quotient ring of \(R\). This theorem gives as a special case the author’s result on differential identities with involution [in Trans. Am. Math. Soc. 316, No. 1, 251-279 (1989; Zbl 0676.16011)].

16R50 Other kinds of identities (generalized polynomial, rational, involution)
16W20 Automorphisms and endomorphisms
16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
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