Nonlinear identities with skew derivations

作者：Chuang Chen Lian^*

来源：Journal of Algebra, 2016, 461: 244-294.

DOI：10.1016/j.jalgebra.2016.05.007

• Prime ring
• Automorphism
• Skew derivation
• Identity
• Coalgebras
• Pointed coalgebras
• The complete ring of quotients
• Utumi quotient ring

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

• 出版日期2016-9-1