Bases for skew derivations

Abstract

Let R be a prime ring, Q the Utumi quotient ring of R and Ω the expansion closed set of products of automorphisms and skew derivations of Q. Let Λ⊆Ω∖Ωm, where m≥0, be such that for any w∈Λ(g,h), where g,h are automorphisms of Q, w(xy)−g(x)w(y)−w(x)h(y) is a sum of terms with δ(x) and δ′(y), where δ∈Λ|w|−1orδ′∈Λ|w|−1orδ,δ′∈Ωm (Definition 3). Suppose that no monic linear identities with words in Λ∪Ωm involve words in Λ. We prove the following:(i) Λ extends to a basis of Ω (Theorem 1).(ii) If a generalized polynomial φ with words in Λ∪Ωm is an identity, then so is the generalized polynomial obtained from φ by replacing every occurrence of ν(x) in φ by an arbitrary new variable yν,x for ν∈Λ and variable x (Theorem 2).Dedekind's Lemma asserts that distinct automorphisms σi of a commutative field are independent maps over the field and hence, via linearization, σi(xj) in distinct variables xj, considered as distinct maps for distinct ordered pair (i,j), are transcendental over the field in the case of characteristic 0. (ii) above generalizes this to maps defined by w∈Λ with dependency and transcendency over the prime ring R.