質環上之導算等式與代數自同構之常值

Abstract

Abstract This thesis focuses on differential identities and constants of algebraic automorphisms in prime rings. In Chapter 1 we prove that an algebra over a field with a finite dimensional maximal subalgebra must be finite dimensional. In Chapters 2 and 3 we consider certain differential identities in prime rings. Firstly, we show that if a prime algebra admits a nonzero generalized skew derivation with algebraic values of bounded degree, then the algebra must be a primitive ring with nonzero socle and its associated division algebra is a finite-dimensional central division algebra. Secondly, we determine the structure of a prime ring admitting an additive n-commuting map which is linear over its center. In Chapter 4 we consider constants of algebraic automorphisms in prime rings. Let R be a prime ring with extended centroid C. For an automorphism sig of R we let R^(sig)≡{x in R | sig(x)=x}, the subring of constants of sig on R. Suppose that the automorphism sig is algebraic over C. We give a complete characterization of the primeness and semiprimeness of the subring R^(sig). Moreover, if the subring R^(sig) is a prime PI-ring, we obtain the PI-degree of R^(sig) in terms of that of the whole ring R and the inner degree of the automorphism sig.