Orthogonal Generalized Derivations In Semiprime Rings And Annihilator On Co-commutators With Derivations In Prime Rings

In 1957, E. C. Posner initiated the study of derivations and centralizing mappings, and gave the well-known Posner Theorem. From then on, there are many papers with different methods extending and perfecting Posner Theorem on prime rings, semiprime rings and their subsets including ideals, one-side ideals, Lie ideals and so on. Generalized polynomial identity theory is an important branch in ring theory. Many profound results have been obtained since the results by V. K. Kharchenko on generalized polynomial identities involving derivations and automorphisms in prime and semiprime rings.This dissertation is devoted to three topics: annihilator on co-commutators with derivations on prime rings and its Lie ideals, orthogonal generalized derivations in semiprime rings, symmetric bi-derivation on semiprime rings and Lie ideal of prime rings.In 2004, E. Albas and N. Argac studied an identity on prime rings: aD(x)= G(x)a. This dissertation studies a general case aD(x)=G(x)b which extends the result of E. Albas and N. Argac. Combining the definition of orthogonal derivations in semiprime rings by M. Bresar and the definition of generalized (θ,(?))-derivation in prime rings by M. Ashraf, the concept orthogonal generalized (θ,(?))-derivations in semiprime rings is given: Let R be a semiprime ring, generalized (θ_i,(?)_i)-derivations are called orthogonal if F₁(x)RF₂(y) — 0 = F₂(y)RF₁(x)for any x, y G R. And the structure of orthogonal generalized (0, ^-derivations are depicted detailedly in this dissertation.Let R be a ring with center Z. A mapping / of R into itself is called centralizing on a subset S of R if [5, f(s)} G Z for all s in S;in the special case, where [s,/(s)] = 0 for all s in S, the mapping / is said to be commuting on S.The question about the linear combination of derivations in prime rings comes from an identity with derivation by M. Bresar. d{x) = ag(x) + h(x)b. There will discuss the question that the linear combination of derivations is commuting on prime rings, and the general expression of derivations will be given in this dissertation;At the same time we extend bi-derivations to this identity, and obtain the result stating under some assumptions that the prime ring is commutative. The study of bi-derivations focus generally on the trace function of bi-additive mappings and there have appeared a lot of related literatures. In 2002, Y. Wang got rid of the condition on the trace function of bi-derivations, and obtained more profound results. This dissertation will develop Wang's results to semiprime rings and Lie ideals in prime rings, such that the questions about bi-derivations are solved more perfectly.In what follows, let R be a prime ring (semiprime ring), Qmr its maximal right quotient ring, Qr its two-sided right quotient ring, Qs its symmetric quotient ring, Z its center, C its extended centroid, RC its central closure, / its nonzero ideal and L its Lie ideal.The main results of this dissertation are as follows:Theorem 2.1.1 Let R be a noncommutative prime ring, d,g two nonzero derivations of R and a G R, Suppose that a(d(xh)xk-xkg(xk)) = 0 for all x G R, where k is a fixed integer. Then a = 0.Theorem 2.2.1 Let R be a prime ring of characteristic different from 2, d and g two derivations of R at least one of which is nonzero, L a non-central Lie ideal of R and a G R. If a(d(u)u - ug(u)) = 0 for any u G L, then either a = 0, or R. is Si-ring, d(x) = \p,x], and g(x) = -d(x) for some p in Qmr the maximal right quotient ring of R.Theorem 3.1.1 Let R be a noncommutative prime ring and (D, a) a generalized derivation of R. If D(xoy) = xoy for all x,y G R, then D is the identical mapping of R.Theorem 3.1.2 Let (D.a) and (G.fi) be two nonzero generalized derivations of prime ring R and a,b G R. If aD(x) = G(x)b for all x G R, then one of the following holds :(i) If b G C and a <ï¿¡ C. Then a = (3 = 0 and there exist p,q G Qr(R.C) such that D(x) — px, G(x) = qx for all x G R;(ii) If a G C and b ï¿¡ C. The same consequence with (i);(in) If a G C and b G C. Then either a = b = 0 or D and G is C-linearly dependent;(w) If a ï¿¡ C and b <ï¿¡ C. Then there exist p,q G Qr(RC) and w 6 Qr(R), such that a(x) = [q,x], fi(x) — [x,p], D(x) — tux — xq and G(x) = xp + avx for all x G R with v G C and aw — pb = 0-Theorem 3.2.1 Let R be a 2-torsion free semiprime ring with generalized (0i, 4>i)-derivations (Fj, di), where 6i, & are subjective endomorphisms on R. Then the following conditions are equivalent:(i) (Fi,di) are orthogonal generalized (8',;,C, C 'â–  R —f C are additive mappings.Theorem 4.2.1 Let R be a 2-torsion free semiprirne ring. If D is a nonzero symmetric bi-derivations of R, then there exists 0 ^ /j 6 C, such that fj,R is commutative.Theorem 4.3.1 Let R be a prime ring of characteristic different from 2, L a Lie ideal of R and D a nonzero symmetric bi-derivations of L, for all u € L, v2 e L. then LC Z{R).Theorem 4.3.2 Let R be a prime ring, D,F,G nonzero bi-derivations of R, a, b 6 R, a $ Z and b $ Z. If D(x,y) = aF(x,y) + G(x,y)b, then R is commutative.