The Power Values Properties Of Generalized Derivations

Ring theory contains generalized polynomial identity (GPI) theory as an important area, which is based upon the investigation of many special identities in prime and semiprime rings. In 1979, Herstein initiated the study of the power values type identities by building two identities (for convenience, call them Herstein type identities): f(x)~n = 0(∈Z),(?)x ∈ S, where f is a map of a ring R, S (?) R, Z denotes the center of R, n is a fixed positive integer. In fact, he gave the solution of Herstein type identities when 5 is a prime ring R and f is a derivation of R. In 1990, Bresar built an identity (for convenience, call it Bresar type identity): af(x)~n = 0,(?)x ∈ S, where a ∈ R. Bresar gave the solution of Bresar type identity when S is an (n— 1)!-torsionfree semiprime ring R and f is a derivation of R. Since 1979, many authors have obtained the solutions of Herstein type identities and Bresar type identity when S is a subset of a prime ring or semiprime ring R (for example, ideal, Lie ideal, one-sided ideal and the image set of a multilinear polynomial under an ideal or one-sided ideal) and f is some map of R (for example, derivation, generalized derivation, (α,β)-derivation, generalized skew derivation). These results are important parts in the study of the power values type identities, and are important materials to GPI theory.This dissertation develops the study of the power values type identities by building three new identities. The results obtained describe the power values properties of generalized derivations in prime rings from different directions.In what follows, let R be a prime ring or a semiprime ring (not necessarily with identity), Z its center, C its extended centroid, U its maximal right ring of quotients, Qr its two-sided right ring of quotients, Qs its symmetric ring of quotients, S a subset of R and / a map of R.Firstly, two new identities (for convenience, call them the condition I type identities and condition II type identities, respectively)are built:/. a{f{x)b)n = 0(6 C), Vx e S,II. {af(qx) - bx)n = 0(€ C), Vx e S,where a, b,q e U. The condition I type identities are the generalization of Herstein type identities and Bresar type identity. The condition II type identities are only the development of Herstein type identities. The author gives the solutions of the condition I, II type identities when S is a prime ring and / is a generalized derivation of R. By the method of orthogonal completions, the semiprime ring cases are obtained. These results generalize the ones of Herstein and Bresar, simultaneously.Secondly, the concept " maps with skew nilpotent values " is given, that is the map satisfying the following identityf{x)1' xmi ■ ■ ■ /(x)**-^"1*-1 f(x)tk = 0, Vx 6 5,where k > 1, ti,...,tk, mi,..., m*_i are positive integers. The above identity is the development of the Herstein nilpotent type identity. The author characters the construction of generalized derivations with skew nilpotent values on a non-commutative Lie ideal of a prime ring. The results obtained generalize the ones of Herstein, Lee and Carini, simultaneously.At last, the author characters the generalized derivation having the same power values with the left multiplication in a noncommutative Lie ideal L and a nonzero left ideal ￡ of a prime rings R. The author believes that two generalized derivations <5i and 82 of a prime ring R must be C-linear dependent if R satisfies the identity (^(x)" = ^(x)". But it is difficult to solve the problem. In order toexplore some methods, the author discusses the problem when 82(x) — ax and L, C satisfy the identity above and obtains two neat results which generalize the ones by Lee and Lanski, simultaneously.The main results of this dissertation are as followsTheorem 2.2.1 Let R be a prime ring, / a nonzero ideal of R,a,be U\{0}, 8 a generalized derivation of R, n a fixed positive integer, then1. If / satisfies (a8(x))nb = 0, then there exist a\,bi G U such that 6(x) = aix + xb\, V.x G R, aa\ = 0 and at least one of b\a, b\b is zero.2. If / satisfies {a6(x))nb G C and (a5(xo))nb ^ 0 for some x0 G /, then R satisfies 54. Particularly if a, b G R\J {1}, then R is either commutative or an order in a 4-dimensional simple algebra.Theorem 2.2.2 Let R be a semiprime ring, a, b G U, 8 a generalized derivation of R, n a fixed positive integer. If R satisfies (a8(x))nb G C, then there are pairwise orthogonal idempotents ei,e2,63 G U such that ej + e.2 + ej — 1, e\V satisfies a5(x)a = 0, e