具Engel條件之導算恆等式

Abstract

This thesis focuses on kernel inclusions of algebraic automorphisms, generalized derivations with Engel conditions and generalized derivations cocentralizing polynomials. In Chapter 1 we consider algebraic automorphisms with kernel inclusions. Let R be a prime ring. For an automorphism σ of R we let R(σ) def. = {x ∈ R | σ(x) = x}. Assume that σ is algebraic. We characterize the automorphism τ of R such that R(σ) ⊆ R(τ). In Chapters 2,3 and 4 we consider certain identities with generalized derivations. Firstly, we concern generalized derivations cocentralizing polynomials. Let R be a prime ring with extended centroid C and let f(X1, . . . ,Xt) be a polynomial over C with zero constant term. Let D and G be generalized derivations of R. We characterize D,G and f(X1, . . . ,Xt) satisfying D(f(x1, . . . , xt))f(x1, . . . , xt) − f(x1, . . . , xt)G(f(x1, . . . , xt))∈ C for all x1, . . . , xt in R. Secondly, we consider certain Engel conditions on polynomials with generalized derivations. Precisely, we characterize D and f(X1, . . . ,Xt) such that the following Engel identity is satisfied: [D(f(x1, . . . , xt)), f(x1, . . . , xt)]k= 0 for all x1, . . . , xt in R. At the end, we concern a generalization of the previous two situations. Precisely, we characterize D,G and f(X1, . . . ,Xt) satisfying [D(f(x1, . . . , xt))f(x1, . . . , xt)−f(x1, . . . , xt)G(f(x1, . . . , xt)), f(x1, . . . , xt)]k= 0 for all x1, . . . , xt in R.