Nonlinear identities with skew derivations
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文摘
Let R be a prime ring with the extended centroid c. Suppose that R is acted by a pointed coalgebra with group-like elements acting as automorphisms of R. A generalized polynomial with variables acted by the coalgebra is called an identity if it vanishes on R. We prove the following:
(1) If c is a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R with variables acted by Frobenius automorphisms.
(2) If c is not a perfect field, then any such identity is a consequence of simple basic identities defined in [6] and GPIs of R.
With this, we extend Yanai's result [25] to “nonlinear identities”. These are actually special instances of our Theorems 1 and 2 below respectively, which extend Kharchenko's theory of differential identities and to the context of expansion closed word sets introduced in [6].