Shirshov finiteness over rings of constants

Abstract

Let L be a finite-dimensional restricted differential Lie C-algebra of R-continuous derivations of a prime ring R of characteristic p>0, with generalized centroid C. We prove that if the associative inner part of L is quasi-Frobenius then R contains a nonzero element a and elements v1,…,vn, such that for any x∈R we have the expansion\(ax = \sum\limits_{i = 1}^n {v_i \tau _i (x)} \), where\(\tau _i \) are homorphisms of right RL-modules\(\tau _i :R \to R^L = \{ r \in R|\forall \mu \in L(r\mu = 0)\} \). This gives rise to a certain relation on a ring over some subring, known as Shirshov local finiteness. The structure of (R, RL)-subbimodules in a left Martindale ring of quotients is elucidated.