Nonlinear identities with skew derivations

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 [14] and [15] to the context of expansion closed word sets introduced in [6].