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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)].

##### MSC:
 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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