Generalized (θ, Φ)-Derivations In Prime And Semiprime Rings

In resent years, generalized polynomial identities (GPIs) in rings have been widely studied. Since the well known paper of Posner appeared in 1957, there is a growing literature on the research of GPIs in rings. Especially in the last two decades, many authors have extended Posner's theorem, and many important results have been achieved.In 1989, Bresar and Vukman introduced the notion of orthogonality for a pair of derivationsd, g of a semiprime ring, and gave several necessary and sufficient conditions for d, g to be orthogonal on a semiprime ring. In 2004, Argac, Nakajima and Albas, introduced the notionof orthogonality for a pair of generalized derivations (D, d), (G, g) of a semiprime ring and gave several equivalent conditions for (D, d), (G, g) to be orthogonal on a semiprime ring. In 2007, Albas, extended the results of Bresar and Vukman to orthogonal generalized derivations on a nonzero ideal of a semiprime ring. Also in 2007, G(?)1basi and Aydin studied (θ,φ)-derivations and generalized (θ,φ)-derivations satisfyingGθ=θG, Gφ=φG, Dθ=θD, Dφ=φD, gθ=θg, gφ=φg, dθ=θd, dφ=φd,whereθ,φare automorphisms, and gave several equivalent conditions for (D, d), (G, g) to be orthogonal on a semiprime ring. In this thesis we shall study the orthogonality of a couple of generalized (θ,φ)-derivations on a nonzero ideal of a semiprime ring.In 1957, a classical result of Herstein asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation. In 1975, Cusack generalized Herstein's result to 2-torsion free semiprime rings. The relationship between Jordan derivation and derivation has been extended to generalized Jordan derivation and generalized derivation by Ashraf and Al-Shammakhin 2002. They proved that any generalized Jordan derivation on a 2-torsion free prime ring is a generalized derivation. In 2003, Jing and Lu conjectured that any generalized Jordan derivation on a 2-torsion free semiprime ring is a generalized derivation. In 2007, a simple proof of the conjecture is given by Vukman. Also in 2004, Ashraf discussed the relationshipbetween generalized Jordan (θ,φ)-derivation and generalized (θ,φ)-derivation on some Lie ideals of a prime ring, and proved that if U is a non-commutative Lie ideal of a 2-torsion free prime ring R such that u~2∈U for all u∈U and D : R→R is a generalized Jordan(θ,φ)-derivation on U withθan isomorphism, then D is a generalized (θ,φ)-derivation on U. In the thesis, we shall discuss the relation between generalized Jordan (θ,φ)-derivation and generalized (θ,φ)-derivation on left ideals of a prime ring.The main results of this thesis are as follows.Theorem 3.1 Let R be a 2-torsion free semiprime ring, and let I be a nonzero ideal such that l(I) = 0. Suppose thatθ,φare automorphisms of R, and (D, d) and (G, g) are generalized (θ,φ)-derivations of R satisfyingGθ=θG, Gφ=φG, Dθ=θD, Dφ=φD, gθ=θg, gφ=φg, dθ=θd, dφ=φd.Then the following consequences are equivalent:(i) (D, d) and (G, g) are orthogonal;(ii) D(x)G(y) + G(x)D(y) = d{x)G(y) + g(x)D(y) = 0 for all x,y∈I;(iii) D(x)G(y) = d(x)G(y) = 0 for all x,y∈I;(iv) D(x)G(y) = 0 for all x,y∈I and dG = dg = 0 on I;(v) (DG,dg) is a generalized (θ~2,φ~2)-derivation from I to R and D(x)G(y) = 0 for all x,y∈I.Theorem 3.2 Let R be a 2-torsion free semiprime ring, and let I be a nonzero ideal such that l(I) = 0. Suppose thatθ,φare automorphisms of R, and (D, d) and (G, g) are generalized (θ,φ)-derivations of R satisfyingGθ=θG, Gφ=φG, Dθ=θD, Dφ=φD, gθ=θg, gφ=φg, dθ=θd, dφ=φd.Then the following consequences are equivalent:(i) (DG, dg) is a generalized (θ~2,φ~2)-derivation from I to R;(ii) (GD, gd) is a generalized (θ~2,φ~2)-derivation from I to R;(iii) D and g are orthogonal, and G and d are orthogonal.Theorem 4.1 Let R be a 2-torsion free prime ring, and let U be a nonzero right (left) ideal such that l(U) = 0 (r(U) = 0). Suppose thatθ,φare endomorphisms of R such thatθis an automorphism. If D : R→R is a generalized Jordan (θ,φ)-derivation on U, then D is a generalized (θ,φ)-derivation on U.Theorem 4.2 Let R be a 2-torsion free ring, and let U be a left (right) ideal. Suppose thatθ,φare endomorphisms of R such thatθis an automorphism. Suppose that U has a commutator which is not a zero divisor. If D : R→R is a generalized Jordan (θ,φ)-derivation on U, then D is a generalized (θ,φ)-derivation on U.Theorem 4.3 Let R be a 2-torsion free ring, and let U be a commutative non-zero left (right) ideal. Suppose thatθis an automorphism of R and U∩Z(R)≠0. If D : R→R is a generalized Jordan (θ,θ)-derivation on U, then D is a generalized (θ,θ)-derivation on U.