Generalized Derivations On Prime Rings And Semiprime Rings

In resent years, the topics on generalized polynomial identities (GPIs) in rings have been widely investigated. Since Posner's well-known paper appeared in 1957, there is a growing literature on the research of generalized functional identities (GFIs) in rings. Especially in the last two decades, the study of GFIs involving derivations in prime rings and semiprime rings has become increasingly important in ring theory, and many important results have been achieved.This dissertation is devoted to three topics on generalized derivations in prime rings and semiprime rings: cocentralizing generalized derivations, strong commutativity-preserving (scp) generalized derivation, and the composition of two generalized derivations.Let R be a prime ring with center Z and S ∈ R. A pair of mappings d and g from R into itself are called cocentralizing on S if d(x)x — xg(x) ∈ Z for all x ∈ S. In 1993, Bresar initiated the study of cocentralizing mappings. From then on many authors have studied cocentralizing derivations on various subsets of prime rings and semiprime rings. In this dissertation we describe generalized derivations of a prime ring which are cocentralizing on ideals, left ideals and noncentral Lie ideals, respectively. The semiprime case is also considered.A map f : R → R is called scp on S if [f{x), f(y)] = [x, y] for all x, y ∈ S. In 1994, Bell and Daif showed that if there is a derivation d in a semiprime ring R which is scp on a nonzero right ideal p, then p ∈ Z. In this dissertationwe characterize generalized derivations of a prime ring which are scp on ideals and right ideals, respectively. The semiprime case is also considered. Also we prove that if a prime ring R has a nontrivial generalized derivation (or a right partial generalized automorphism) which is scp on a nonzero ideal, then R is commutative.In 1957, Posner showed that in a prime ring with chari? ^ 2 the composition of two nonzero derivations cannot be a derivation of R. In 2004, Lanski proved that in some cases the composition of two nonzero derivations can act like a derivation on a left ideal of prime rings. In 2001, it was shown by T. K. Lee that the composition of some special pair of generalized derivations on a prime ring can be a generalized derivation. In this dissertation we present all possibilities that the composition of two generalized derivations of a prime ring can act like a generalized derivation on a nonzero left ideal. We also obtain that in a prime ring R not satisfying s4, if the composition of two generalized derivations sends a noncentral Lie ideal to zero then the composition must be zero on the whole ring.In what follows, let R be a semiprime ring (not necessarily with identity), Z its center, C its extended centroid, Qmr its maximal right quotient ring and Qs its symmetric quotient ring.The main results of this dissertation are as follows.Theorem 2.1.1 Let R be a noncommutative prime ring with a nonzero ideal /. Suppose that D and G are generalized derivations of R such that D(x)x — xG(x) G Z for all x ￡ /. Then there exists some a G Qmr such that D(x) = xa and G(x) — ax for all x G R.Theorem 2.1.2 Let R be a noncommutative prime ring with a nonzero left ideal A. Suppose that D and G are generalized derivations of R so that D{x)x — xG(x) G Z for all x G A. Then there are a.p G Qmr, q G Qs such that D(x) = xa, G(x) — px + xq for all x G R, and Ag = A (a — p) = 0.Theorem 2.2.1 Let R be a prime ring with a noncentral Lie ideal L. Suppose that D and G are generalized derivations of R so that D(x)x — xG(x) G Zfor all x G L. Then R satisfies S4 or there exists a G QmT such that D{x) — xa and G(x) = ax for all x €. R.Theorem 2.3.1 Let R be a semiprime ring and let D(x) = xa + d(x) and G(x) = bx + g(x) be generalized derivations of R with a, b G Qmr and derivations d, 5. If D and G satisfy D(x)x - xG(x) G Z for all z G A. Then QmrX{a - b) is a central ideal of Qmr, and the ideals (Ad), (i?dA), (Aftd), (AA?) are all contained in Z. If in addition the right annihilator of A is zero, then the ideals (Rd), (R9) C Z.Theorem 3.1.1 Let R be a noncommutative semiprime ring and let D(x) = xa + 5(x) be a generalized derivation of R with a € Qmr and derivation 5. If D is scp on a nonzero right ideal p. Then either R contains a central ideal, or S(p)p = 0.Theorem 3.1.3 Let R be a noncommutative prime ring and let D be a generalized derivation of R which is scp on a nonzero ideal /. Then D(x) = x for all x e R or D(x) = -x for all x € R.Theorem 3.2.1 Let R be a noncommutative prime ring and let D be a generalized derivation of R which is scp on a nonzero right ideal p. Then there are a G Qs, b G Qmr such that D(x) — ax + xb for all x 3, b is a invertible element of End(VF), and V has a basis {w} U {wa}aeA such that 6u>Q = cav + cfwQ, 6w = d~lv for some 0 7^ of, cQ G F, and ew = w, e?;a = 0, where e is an idempotent contained in EndiVp) such that pC C eEnd(VF).Theorem 3.3.1 Let i? be a noncommutative prime ring and let T be a right partial generalized automorphism of R which is scp on a nonzero ideal /. Then T(x) = x for all x G R or T(.x) = -x for all x e R.Theorem 4.2.1 Let R be a prime ring and let D and G be generalized derivations of R such that DG acts like a generalized derivation on a nonzero left...