導算之常值與導算恆等式

Abstract

We show two results in this thesis. Firstly, let R be a semiprime ring with extended centroid C and with Martindale left ring of quotients R_F. Suppose that δ : R → R is a left R_F-integral derivation. Let R(δ) (resp. R_F(δ)) denote the subring of constants of δ on R (resp. R_F). We prove: (I) If the R_F-integral degree of δ is m, then δ is C-integral of degree less than or equal to m^2. (II) R(δ) and R_F(δ) satisfy the same PIs. Secondly, let R be a prime ring with extended centroid C and let f(X_1, . . . ,X_t) be a polynomial over C, which is not central-valued on RC. Let g be a generalized derivation of R, which is not of the form x : R → λx for some λ in C. Suppose that [g(f(x_1, . . . , x_t)), f(x_1, . . . , x_t)] in C for all x_i in R. Then one of the following two cases holds except when charR = 2 and dim_C RC = 4: (1) g(x) = λx + d(x) for some λ in C, where d is an X-outer derivation of R, charR = 2 and f(X_1, . . . ,X_t)2 is central-valued on RC. (2) g(x) = ax + x(a + β) for some a in R_F and some β in C, and f(X_1, . . . ,X_t)2 is central-valued on RC.